Felix' Math Place (Posts about tensor product.)
https://math.fontein.de/tag/tensor-product.atom
2019-11-17T10:38:22Z
Felix Fontein
Nikola
Homomorphisms, Tensor Products and Certain Canonical Maps.
https://math.fontein.de/2010/01/29/homomorphisms-tensor-products-and-certain-canonical-maps/
2010-01-29T07:20:57+01:00
2010-01-29T07:20:57+01:00
Felix Fontein
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> be vector spaces over a field <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/xhMRUT21WeTtJ7b0fU5uoQdBZWQNNqo8JB5nOw.svgz" alt="V^* = \Hom_K(V, K)" title="V^* = \Hom_K(V, K)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="18" src="https://math.fontein.de/formulae/JDWgy_EHLuEnu7LsHsoOqCX0L5IxV2Hpu.Zkog.svgz" alt="W^* = \Hom_K(W, K)" title="W^* = \Hom_K(W, K)"></span> their duals. In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> is finite dimensional, one obtains a non-canonical isomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="13" src="https://math.fontein.de/formulae/r4sxli.DTPUjKSUpwt31acNnQd.EOXvF7PoDxg.svgz" alt="V \cong V^*" title="V \cong V^*"></span>, a canonical isomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="13" src="https://math.fontein.de/formulae/LKwFNgCiNUrM3joRJ9hKp8aVGeLVgj3LgGWIHg.svgz" alt="V \cong V^{**}" title="V \cong V^{**}"></span> and a canonical isomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="201" height="18" src="https://math.fontein.de/formulae/qRICuzpEjEUiRodr6yqMm_AdYWs81awt87No.A.svgz" alt="W^* \tensor_K V \cong \Hom_K(W, V)" title="W^* \tensor_K V \cong \Hom_K(W, V)"></span>.
</p>
<p>
In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="102" height="15" src="https://math.fontein.de/formulae/4cQneuw.0kYEXwV.bXgT22yP3uy5.Vz6pZ75Eg.svgz" alt="\dim_K V = \infty" title="\dim_K V = \infty"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> are not isomorphic: a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> has a cardinality strictly larger than the one of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>. Moreover, the canonical map <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="12" src="https://math.fontein.de/formulae/HX_CAccLs.zyrV9Z37Exosfb.tFmfjm1fe0_1Q.svgz" alt="V \to V^{**}" title="V \to V^{**}"></span> is still a monomorphism, but no longer surjective. In the case of <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="15" src="https://math.fontein.de/formulae/QxnOKbzAk0se4W6KxGki9RpTJPqy.68Zg_Io7A.svgz" alt="V \tensor_K V^*" title="V \tensor_K V^*"></span>, one has as well a canonical monomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="18" src="https://math.fontein.de/formulae/HlAcwyukaAE1bQ7uvuGPElf_DJnz0nk_b9MGFA.svgz" alt="V \tensor_K V^* \to \Hom_K(V, V)" title="V \tensor_K V^* \to \Hom_K(V, V)"></span>, but it is no longer surjective as well. We want to study the images of the canonical maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="12" src="https://math.fontein.de/formulae/HX_CAccLs.zyrV9Z37Exosfb.tFmfjm1fe0_1Q.svgz" alt="V \to V^{**}" title="V \to V^{**}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="18" src="https://math.fontein.de/formulae/HlAcwyukaAE1bQ7uvuGPElf_DJnz0nk_b9MGFA.svgz" alt="V \tensor_K V^* \to \Hom_K(V, V)" title="V \tensor_K V^* \to \Hom_K(V, V)"></span>.
</p>
<p>
We begin with an auxiliary lemma.
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
<a name="nonzeroform"></a>
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/pHE1isivXn2PHBT0jHPa2L8fh.xY.Mg8eTf0dA.svgz" alt="v \neq 0" title="v \neq 0"></span>. Then there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="16" src="https://math.fontein.de/formulae/JzX7ujV9Wgp4Cq2D5Dz8QPEExqAnIWdnFxqXHg.svgz" alt="\varphi \in V^*" title="\varphi \in V^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="18" src="https://math.fontein.de/formulae/bnhFzjVYxj6ueQISRhDhCFKsPXszx7.x_mswdA.svgz" alt="\varphi(v) = 1" title="\varphi(v) = 1"></span>. Hence, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="18" src="https://math.fontein.de/formulae/D.g8nCeKGylCSH_m66TT5XyEgDZjdbBMYd9S8g.svgz" alt="\varphi(v) = 0" title="\varphi(v) = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="16" src="https://math.fontein.de/formulae/JzX7ujV9Wgp4Cq2D5Dz8QPEExqAnIWdnFxqXHg.svgz" alt="\varphi \in V^*" title="\varphi \in V^*"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="11" src="https://math.fontein.de/formulae/.HQUzFhr5L1LgNSwtQMPqO1owFKtqkbZ2QX_Gw.svgz" alt="v = 0" title="v = 0"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Choose a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-basis <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/Tc8aPDKM3vEiPXTaEoqzjFpI17Xa7Hv3utnPpw.svgz" alt="(v_i)_{i \in I}" title="(v_i)_{i \in I}"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> such that there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="37" height="13" src="https://math.fontein.de/formulae/KTLN00XejVK9clqzT1D2R_ihjy99JmrUXAK4xA.svgz" alt="t \in I" title="t \in I"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="10" src="https://math.fontein.de/formulae/XuPg45jLerjhoj6YdDO5kXFo6nTU.kBIPGW23Q.svgz" alt="v_t = v" title="v_t = v"></span>. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="89" height="15" src="https://math.fontein.de/formulae/N3Jff7F8rdDIaOJX7_LlrzUSx_q1QC6sHwr8oQ.svgz" alt="\pi_t : V \to K" title="\pi_t : V \to K"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="20" src="https://math.fontein.de/formulae/m7CUzwBekblqDH36WcKTFDOJYbFiNiery4baRw.svgz" alt="\sum_{i \in I} \lambda_i v_i \mapsto \lambda_t" title="\sum_{i \in I} \lambda_i v_i \mapsto \lambda_t"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="140" height="18" src="https://math.fontein.de/formulae/KA_C90vbOmjZhWWcJR3JdTqy._hkQ57UR6lJEA.svgz" alt="\pi_t(v) = \pi_t(v_t) = 1" title="\pi_t(v) = \pi_t(v_t) = 1"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="15" src="https://math.fontein.de/formulae/uZJMDEovf5Y08ddxxN9GoMvxJ7z9QSFxG2uB5w.svgz" alt="\pi_t \in V^*" title="\pi_t \in V^*"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
<a name="Psimapprop"></a>
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="288" height="53" src="https://math.fontein.de/formulae/5qXlmnpga8eWK9aTcAq.vI6.78FK6OmKsWhasw.svgz" alt="\Psi : V \to V^{**}, \qquad v \mapsto \begin{cases} V^* \to K, \\ \alpha \mapsto \alpha(v) \end{cases}" title="\Psi : V \to V^{**}, \qquad v \mapsto \begin{cases} V^* \to K, \\ \alpha \mapsto \alpha(v) \end{cases}">
</div>
<p>
is a monomorphism and its image is
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="289" height="51" src="https://math.fontein.de/formulae/vz5AlL8P..1V4IBHPkRWblSgZi4klT9YmMTJCg.svgz" alt="\biggl\{ \varphi \in V^{**} \;\biggm|\; \bigcap_{\alpha \in \ker \varphi} \ker \alpha \neq 0 \biggr\} \cup \{ 0 \}." title="\biggl\{ \varphi \in V^{**} \;\biggm|\; \bigcap_{\alpha \in \ker \varphi} \ker \alpha \neq 0 \biggr\} \cup \{ 0 \}.">
</div>
<p>
In particular, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="138" height="21" src="https://math.fontein.de/formulae/HQA0m9NJKOHx0RRROlRffUUxi_rqIR2N0wjfNQ.svgz" alt="\bigcap_{\alpha \in \ker \varphi} \ker \alpha \neq 0" title="\bigcap_{\alpha \in \ker \varphi} \ker \alpha \neq 0"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="16" src="https://math.fontein.de/formulae/JuckGYuLFpwHWmm9.m9mwRA5MmR4Ga4cH70t2A.svgz" alt="\varphi \in V^{**}" title="\varphi \in V^{**}"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="21" src="https://math.fontein.de/formulae/eL5ZoOmVCHQI2Mzw1Gk6waiPwz9Pr0qo1sEcwQ.svgz" alt="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha = 1" title="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha = 1"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Clearly, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="118" height="18" src="https://math.fontein.de/formulae/W2oMP0aWpLhkvfXuYUlw3epAV22JdhDJNcYbpA.svgz" alt="\Psi(v) : V^* \to K" title="\Psi(v) : V^* \to K"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear. Moreover, one quickly sees that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear itself. To see that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is injective, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/pHE1isivXn2PHBT0jHPa2L8fh.xY.Mg8eTf0dA.svgz" alt="v \neq 0" title="v \neq 0"></span>. Now, by the lemma, there exists a <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="13" src="https://math.fontein.de/formulae/B5EZ4M1U6k.W_J9uOkkyIQ4U1qEhUXRFxvRYrA.svgz" alt="\pi \in V^*" title="\pi \in V^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/b4KAmah.Y6S5KEDLHGgKffEN5gHpmoJUxsEkTA.svgz" alt="\pi(v) = 1" title="\pi(v) = 1"></span>; this shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/9291q.f1lj2O6wAkdYTeFRW.gtEF5V0iTe6f4Q.svgz" alt="\Psi(v)(\pi_t) = 1" title="\Psi(v)(\pi_t) = 1"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="18" src="https://math.fontein.de/formulae/HnvEUBbHGCcl5fEIyFbHNGcs2a5DMFeD7YE50A.svgz" alt="\Psi(v) \neq 0" title="\Psi(v) \neq 0"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="12" src="https://math.fontein.de/formulae/xN.rud2.SpOb.RZblWkS0ygaBUoPCFkM1KiYrQ.svgz" alt="\ker \Psi = 0" title="\ker \Psi = 0"></span> an <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is injective.
</p>
<p>
Now, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="96" height="18" src="https://math.fontein.de/formulae/fXaBf99ziyNg52ZSv.q2C6.l8937ulGD6oLuTA.svgz" alt="\alpha \in \ker \Psi(v)" title="\alpha \in \ker \Psi(v)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="152" height="18" src="https://math.fontein.de/formulae/hLlXoWmjzer.s0RNgDeBY3GH6taOs2Biou3Mbg.svgz" alt="\alpha(v) = \Psi(v)(\alpha) = 0" title="\alpha(v) = \Psi(v)(\alpha) = 0"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="157" height="22" src="https://math.fontein.de/formulae/M.0D.CJX17C8TlJc1kXSnbfra7MjgYfa7ovGmw.svgz" alt="v \in \bigcap_{\alpha \in \ker \Psi(v)} \ker \alpha" title="v \in \bigcap_{\alpha \in \ker \Psi(v)} \ker \alpha"></span>. This shows that the image of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is contained in the given set. Now assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="106" height="18" src="https://math.fontein.de/formulae/5pLxo6PML4g9xb4oOz6BcQ4LzVoVOEDPGZxvew.svgz" alt="\varphi \in V^{**} \setminus \{ 0 \}" title="\varphi \in V^{**} \setminus \{ 0 \}"></span> satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="138" height="21" src="https://math.fontein.de/formulae/Kj_1r1kZw_FeKUg25TCwmVWr38gDH11pxf1TFA.svgz" alt="\bigcap_{\alpha \in \ker\varphi} \ker \alpha \neq 0" title="\bigcap_{\alpha \in \ker\varphi} \ker \alpha \neq 0"></span>; say, <span class="inline-formula"><img class="img-inline-formula img-formula" width="180" height="21" src="https://math.fontein.de/formulae/HLwB_oz7uS7tp_62QneX_x3RrSU8cz9RqGyIow.svgz" alt="v \in \bigcap_{\alpha \in \ker\varphi} \ker \alpha \setminus \{ 0 \}" title="v \in \bigcap_{\alpha \in \ker\varphi} \ker \alpha \setminus \{ 0 \}"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="18" src="https://math.fontein.de/formulae/mt0S0BuYrPcnPijQ9PI2fvS6ptCae4arqS0NZw.svgz" alt="\alpha(v) = 0" title="\alpha(v) = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="16" src="https://math.fontein.de/formulae/9BzjkcL7pBLzPO0nmLj7cFveDMb3jPnlvSGKwA.svgz" alt="\alpha \in \ker\varphi" title="\alpha \in \ker\varphi"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="125" height="18" src="https://math.fontein.de/formulae/7VEv5hqRpwcL2Un7DZwrfu7w8dTCTP4aTEYAKw.svgz" alt="\ker \varphi \subseteq \ker \Psi(v)" title="\ker \varphi \subseteq \ker \Psi(v)"></span>. By the Homomorphism Theorem, there exists a homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="143" height="18" src="https://math.fontein.de/formulae/oz7.24nxcKxry9e4n2JttcHB0f4cSUNmwyErVw.svgz" alt="\tilde{\varphi} : V^* / \ker \varphi \to K" title="\tilde{\varphi} : V^* / \ker \varphi \to K"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="210" height="105" src="https://math.fontein.de/formulae/lpiw1YD_YN6Se_dPybC5byf75xmvy0bA3JSOyA.svgz" alt="\xymatrix{ V \ar[rr]^{\Psi(v)} \ar[dr]_{\pi} & & K \\ & V / \ker \varphi \ar[ru]_{\tilde{\varphi}} & }" title="\xymatrix{ V \ar[rr]^{\Psi(v)} \ar[dr]_{\pi} & & K \\ & V / \ker \varphi \ar[ru]_{\tilde{\varphi}} & }">
</div>
<p>
commutes. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="176" height="18" src="https://math.fontein.de/formulae/Hxc3UKmHxF39DvQdu8Fc96NMh_mxErg4JgxZMQ.svgz" alt="V^* / \ker \varphi \cong \varphi(V) = K" title="V^* / \ker \varphi \cong \varphi(V) = K"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="151" height="18" src="https://math.fontein.de/formulae/sKEqeNYXr6Z5rkQL58al_0yNwWnYzXKjYc36Tw.svgz" alt="\dim_K V^* / \ker \varphi = 1" title="\dim_K V^* / \ker \varphi = 1"></span>. As <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="16" src="https://math.fontein.de/formulae/uNcK_tL1Xp7EV7P67OgE9kXgWEoZkrfc_6bx6w.svgz" alt="\tilde{\varphi} \neq 0" title="\tilde{\varphi} \neq 0"></span> (as <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/pHE1isivXn2PHBT0jHPa2L8fh.xY.Mg8eTf0dA.svgz" alt="v \neq 0" title="v \neq 0"></span>), <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="15" src="https://math.fontein.de/formulae/Kf6xDpQPp2pRij4BkZcvPyQo6CGHbfeQ727fgQ.svgz" alt="\tilde{\varphi}" title="\tilde{\varphi}"></span> is an isomorphism and we must have <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/NdWS_rgUUoUr8i_Cmh6Najdl9DKtgGN.erQx0Q.svgz" alt="\Psi(v) = \lambda \varphi" title="\Psi(v) = \lambda \varphi"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="18" src="https://math.fontein.de/formulae/yQ4Vw9DY35YUq3DWRihCr0eZXI1xxDo8ZFjwrw.svgz" alt="\lambda \in K \setminus \{ 0 \}" title="\lambda \in K \setminus \{ 0 \}"></span>. But then, <span class="inline-formula"><img class="img-inline-formula img-formula" width="101" height="19" src="https://math.fontein.de/formulae/7NKb7JIVYI1Fm9R9KK8iVOs6XC3cy1sjpMQCsQ.svgz" alt="\varphi = \Psi(\lambda^{-1} v)" title="\varphi = \Psi(\lambda^{-1} v)"></span> lies in the image of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span>.
</p>
<p>
Finally, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="21" src="https://math.fontein.de/formulae/VNBWywZzHL0ipYP1ZYINlQyRSk7e0WWX1cnHMg.svgz" alt="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha > 0" title="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha > 0"></span>, we saw that we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="90" height="18" src="https://math.fontein.de/formulae/M_Gw3ggyNU3Xpoy1pMWliD41yD5NptIMrIpzVg.svgz" alt="\varphi = \lambda_v \Phi(v)" title="\varphi = \lambda_v \Phi(v)"></span> for any non-zero <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="21" src="https://math.fontein.de/formulae/6FGLt1.kKinSf7PhhjODC6rhVYhFEqzBP8vxdg.svgz" alt="v \in \bigcap_{\alpha \in \ker \varphi} \ker \alpha" title="v \in \bigcap_{\alpha \in \ker \varphi} \ker \alpha"></span>, with <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/2ICgsylmlH81L56dY5JENVn1khEDt6ra5Ahbpw.svgz" alt="\lambda_v \in K \setminus \{ 0 \}" title="\lambda_v \in K \setminus \{ 0 \}"></span> depending on <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/KsOArZbMnsJRrzwj8HWDL_L45lvecz0OO6uAFA.svgz" alt="v" title="v"></span>. Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="12" src="https://math.fontein.de/formulae/_MI6A1fftSkWBJe.edvswzMjX8.IkzyS0iL96g.svgz" alt="\Phi : V \to V^{**}" title="\Phi : V \to V^{**}"></span> is injective, this shows that we must have <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="21" src="https://math.fontein.de/formulae/eL5ZoOmVCHQI2Mzw1Gk6waiPwz9Pr0qo1sEcwQ.svgz" alt="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha = 1" title="\dim_K \bigcap_{\alpha \in \ker \varphi} \ker \alpha = 1"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
This allows us to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="12" src="https://math.fontein.de/formulae/HX_CAccLs.zyrV9Z37Exosfb.tFmfjm1fe0_1Q.svgz" alt="V \to V^{**}" title="V \to V^{**}"></span> is surjective if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="13" src="https://math.fontein.de/formulae/H2t7THeYUYfwyxjytGm7sfIMRiypTCKRWgz_fA.svgz" alt="\dim V < \infty" title="\dim V < \infty"></span>.
</p>
<div class="theorem-environment theorem-corollary-environment">
<div class="theorem-header theorem-corollary-header">
Corollary.
</div>
<div class="theorem-content theorem-corollary-content">
<p>
We have that <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="12" src="https://math.fontein.de/formulae/S.5reg1mC6yDupvYuUQf3m2oWl8kBCuZzuXXyQ.svgz" alt="\Psi : V \to V^{**}" title="\Psi : V \to V^{**}"></span> is surjective if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="13" src="https://math.fontein.de/formulae/H2t7THeYUYfwyxjytGm7sfIMRiypTCKRWgz_fA.svgz" alt="\dim V < \infty" title="\dim V < \infty"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
First, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="13" src="https://math.fontein.de/formulae/H2t7THeYUYfwyxjytGm7sfIMRiypTCKRWgz_fA.svgz" alt="\dim V < \infty" title="\dim V < \infty"></span>, we see that <span class="inline-formula"><img class="img-inline-formula img-formula" width="211" height="12" src="https://math.fontein.de/formulae/0nIJMlrSn4pWlGIKq2KjRtRhz9_Rbg2b6.FCdA.svgz" alt="\dim V^{**} = \dim V^* = \dim V" title="\dim V^{**} = \dim V^* = \dim V"></span>. Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is injective, it follows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> is in fact an isomorphism.
</p>
<p>
Now assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="12" src="https://math.fontein.de/formulae/GNg0tcV4Zy7s4EVLSObp6kS0zMwCtZIn0j48fQ.svgz" alt="\dim V = \infty" title="\dim V = \infty"></span>. It suffices to construct a hyperplane <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> in <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="120" height="20" src="https://math.fontein.de/formulae/.Cu_NDhnxfPnHMjgi.CxMBFkduYi3vZSI.K3PA.svgz" alt="\bigcap_{\alpha \in H} \ker \alpha = 0" title="\bigcap_{\alpha \in H} \ker \alpha = 0"></span>; this defines an element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="30" height="12" src="https://math.fontein.de/formulae/ve2bmaH_8chhZvCkWF1z5C7EZZTpNh6JYjbz0g.svgz" alt="V^{**}" title="V^{**}"></span> which is not contained in the image of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span> by the <a href="https://math.fontein.de/2010/01/29/homomorphisms-tensor-products-and-certain-canonical-maps/#Psimapprop">above proposition</a>. For that, chose a basis <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/oj9Aog9a3HJ5YRMYgI6v52UDUjEka9FUI_UbfA.svgz" alt="(v_i)_{i\in I}" title="(v_i)_{i\in I}"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> (using <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">Zorn's lemma</a>). This defines a family of elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="89" height="15" src="https://math.fontein.de/formulae/cw6Arn7UCtCP5gSV.gRzmVHHyW7w4MOGV3PDdg.svgz" alt="\pi_i : V \to K" title="\pi_i : V \to K"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="21" src="https://math.fontein.de/formulae/98qKjLRCafVQyyyhlHrb5PVIHfPNssbsCZcYOw.svgz" alt="\sum_{j\in I} \lambda_j v_j \mapsto \lambda_i" title="\sum_{j\in I} \lambda_j v_j \mapsto \lambda_i"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="13" src="https://math.fontein.de/formulae/_1LQK7goZihMcFOa3BVLgigmVredVdZ1ubh62g.svgz" alt="H'" title="H'"></span> be the subspace of <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> generated by the <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="10" src="https://math.fontein.de/formulae/Z_Dw51C9uY3GxsXD_CmEzfMR8V0d8mF17x6KVg.svgz" alt="\pi_i" title="\pi_i"></span>'s. If we would have <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="19" src="https://math.fontein.de/formulae/oxFNgHsl7us7YO90.IaN0cubWf77rNmMZq8fmg.svgz" alt="H' \subsetneqq V^*" title="H' \subsetneqq V^*"></span>, we could emply Zorn's lemma a second time to obtain a hyperplane <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="15" src="https://math.fontein.de/formulae/R8N9F9lszypbO1dmXl0ahmQLM_7cecVFEaH81Q.svgz" alt="H \subseteq V^*" title="H \subseteq V^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="16" src="https://math.fontein.de/formulae/AgBuM2nA04BnQLDRbXWhpdjSdHUbRVIKPUlrFg.svgz" alt="H' \subseteq H" title="H' \subseteq H"></span>; this would prove our claim.
</p>
<p>
Hence, we have to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="17" src="https://math.fontein.de/formulae/Ok1o3iqB1XZmWdejlvHrIQBPLhWbRJ6lQTsEQg.svgz" alt="H' \neq V^*" title="H' \neq V^*"></span>. Note that for <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="20" src="https://math.fontein.de/formulae/Knexgizg0q9NfXvqKt99DwywaYgTck1WHekWnQ.svgz" alt="v = \sum_{i\in I} \pi_i(v) v_i" title="v = \sum_{i\in I} \pi_i(v) v_i"></span>; in particular, for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, only finitely many of the <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="18" src="https://math.fontein.de/formulae/SnATlkvGbVEvQf8Tg9s5ACb5Pm2gKjAi_alAbA.svgz" alt="\pi_i(v)" title="\pi_i(v)"></span>'s are non-zero. Hence, it makes sense to define <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="12" src="https://math.fontein.de/formulae/TVu52l30928laRP73XumpvLlQznBzwPJo6ZH3Q.svgz" alt="\pi : V \to K" title="\pi : V \to K"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="120" height="20" src="https://math.fontein.de/formulae/muJnl5PG5yqkunwTCu1e0SN0YS9JNR3FdsDLKw.svgz" alt="v \mapsto \sum_{i\in I} \pi_i(v)" title="v \mapsto \sum_{i\in I} \pi_i(v)"></span>. We claim that <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="17" src="https://math.fontein.de/formulae/Oj0Px.XeNZbFU._dePqhRBETyaKYKkxlFFZd9w.svgz" alt="\pi \not\in H'" title="\pi \not\in H'"></span> in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="18" src="https://math.fontein.de/formulae/YGSTRzVBiD.pLjkXZMMfyhR1Rd0WIO5sK6PwDQ.svgz" alt="\abs{I} = \infty" title="\abs{I} = \infty"></span>: for that, note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="18" src="https://math.fontein.de/formulae/1X_mUU1LJ_Yxzde5V3RLmDZrN9d1Cvpql4bLBw.svgz" alt="\{ \pi_i \}_{i\in I}" title="\{ \pi_i \}_{i\in I}"></span> is a linear independent set in <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span>, since for every linear combination <span class="inline-formula"><img class="img-inline-formula img-formula" width="130" height="18" src="https://math.fontein.de/formulae/37xCN2yPIlWdwCE8H5xIu_cTwI791y4.b16biQ.svgz" alt="\sum \lambda_i \pi_i = 0 \in V^*" title="\sum \lambda_i \pi_i = 0 \in V^*"></span>, we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="174" height="21" src="https://math.fontein.de/formulae/viXWShK8gvwXpA3cT_enPx8Rbw3j3F8vEcRzjw.svgz" alt="0 = \bigl(\sum \lambda_i \pi_i \bigr)(v_j) = \lambda_j" title="0 = \bigl(\sum \lambda_i \pi_i \bigr)(v_j) = \lambda_j"></span> for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="16" src="https://math.fontein.de/formulae/22vJfgnD9A6.dKr6S9A8NOqYGu_6zVX5gqNXTw.svgz" alt="j \in I" title="j \in I"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Note that in fact, the proof shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/ggBNCUqRJMnUEd2xA5ta5YI2aAE60fJsFCtMvw.svgz" alt="V^*" title="V^*"></span> is isomorphic to a <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="12" src="https://math.fontein.de/formulae/CboJEamK5MBgQXaRM8xkzPP6yK3NLsAUH1CNJw.svgz" alt="\dim V" title="\dim V"></span>-fold direct product of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, while <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> is isomorphic to a <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="12" src="https://math.fontein.de/formulae/CboJEamK5MBgQXaRM8xkzPP6yK3NLsAUH1CNJw.svgz" alt="\dim V" title="\dim V"></span>-fold direct sum of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>. In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="13" src="https://math.fontein.de/formulae/H2t7THeYUYfwyxjytGm7sfIMRiypTCKRWgz_fA.svgz" alt="\dim V < \infty" title="\dim V < \infty"></span>, these are of the same dimension, but in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="12" src="https://math.fontein.de/formulae/GNg0tcV4Zy7s4EVLSObp6kS0zMwCtZIn0j48fQ.svgz" alt="\dim V = \infty" title="\dim V = \infty"></span>, they are not.
</p>
<p>
We continue with the canonical map <span class="inline-formula"><img class="img-inline-formula img-formula" width="205" height="18" src="https://math.fontein.de/formulae/ABIExN_BXnESIOIBK5fzbo8Kv.7b_BlGDnIPHQ.svgz" alt="W^* \tensor_K V \to \Hom_K(W, V)" title="W^* \tensor_K V \to \Hom_K(W, V)"></span>.
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="466" height="53" src="https://math.fontein.de/formulae/Xte7SFm12zgbkLQa9BN9mSncdAp7yB_AB0UoaQ.svgz" alt="\Phi : W^* \tensor_K V \to \Hom_K(W, V), \qquad \alpha \tensor v \mapsto \begin{cases} W \to V, \\ w \mapsto \alpha(w) v \end{cases}" title="\Phi : W^* \tensor_K V \to \Hom_K(W, V), \qquad \alpha \tensor v \mapsto \begin{cases} W \to V, \\ w \mapsto \alpha(w) v \end{cases}">
</div>
<p>
is a monomorphism and its image is
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="449" height="22" src="https://math.fontein.de/formulae/XjThbMQoJERMNO1t9jq_tvHG6H0LCoFYz2QftA.svgz" alt="\Hom_K^{fin}(W, V) := \{ \varphi \in \Hom_K(W, V) \mid \dim_K \varphi(W) < \infty \}," title="\Hom_K^{fin}(W, V) := \{ \varphi \in \Hom_K(W, V) \mid \dim_K \varphi(W) < \infty \},">
</div>
<p>
the <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector space of finite dimensional <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-homomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="12" src="https://math.fontein.de/formulae/ay4ArWZQ44V_Rc0fYZZ6mtK8OkBf9siuqqI3Eg.svgz" alt="W \to V" title="W \to V"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
One quickly sees that <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="18" src="https://math.fontein.de/formulae/sRQKDi3ByORBJzg8T4Jd4dD5z45DXcswkX3YPw.svgz" alt="w \mapsto \varphi(w) v" title="w \mapsto \varphi(w) v"></span> defines an element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="22" src="https://math.fontein.de/formulae/gl9bCh5i0BN0fKwaBO5eyEHQyEYuSllJQ85cjQ.svgz" alt="\Hom_K^{fin}(W, V)" title="\Hom_K^{fin}(W, V)"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/h1YPUpOWbsd4f48ktllbLrRQbkfVpcJAOVaMmA.svgz" alt="\Phi" title="\Phi"></span> is well-defined and its image is contained in <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="22" src="https://math.fontein.de/formulae/gl9bCh5i0BN0fKwaBO5eyEHQyEYuSllJQ85cjQ.svgz" alt="\Hom_K^{fin}(W, V)" title="\Hom_K^{fin}(W, V)"></span>. Moreover, one quickly sees that <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/h1YPUpOWbsd4f48ktllbLrRQbkfVpcJAOVaMmA.svgz" alt="\Phi" title="\Phi"></span> is a homomorphism.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="214" height="20" src="https://math.fontein.de/formulae/1SLvysS4siG43rZfp8wHlZ68mMxPm9f.S7ZmrA.svgz" alt="x = \sum_{i=1}^n \alpha_i \tensor v_i \in W^* \tensor V" title="x = \sum_{i=1}^n \alpha_i \tensor v_i \in W^* \tensor V"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="18" src="https://math.fontein.de/formulae/O4HHahMtdCVFyD5Vy1qltSh4hhMBri6qgIeQtQ.svgz" alt="\Phi(x) = 0" title="\Phi(x) = 0"></span>, i.e. with <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="20" src="https://math.fontein.de/formulae/uNpArNwinu2UBt.TVUYjBklscthS.EUSi6bnAQ.svgz" alt="\sum_{i=1}^n \alpha_i(w) v_i = 0" title="\sum_{i=1}^n \alpha_i(w) v_i = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="13" src="https://math.fontein.de/formulae/t6ueWtWakzOc_bRQuGKpq7NWCrXp9Boi3AQWsQ.svgz" alt="w \in W" title="w \in W"></span>. Without loss of generality, we can assume that our representation of <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="8" src="https://math.fontein.de/formulae/VCj4le645BgRKl_EPkn4ejR4nVs2ZZW2.MXY9Q.svgz" alt="x" title="x"></span> satisfies that the <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="10" src="https://math.fontein.de/formulae/SxCwN324nn_6_0W3icDz3uxgjoPTEryTXAcDUA.svgz" alt="v_i" title="v_i"></span>'s are linearly independent. In that case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="20" src="https://math.fontein.de/formulae/uNpArNwinu2UBt.TVUYjBklscthS.EUSi6bnAQ.svgz" alt="\sum_{i=1}^n \alpha_i(w) v_i = 0" title="\sum_{i=1}^n \alpha_i(w) v_i = 0"></span> implies <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/msU4IZJR9.7bDy93P4iNxgDCbBl0SQJ.6jxG3A.svgz" alt="\alpha_i(w) = 0" title="\alpha_i(w) = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. But since this is true for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="13" src="https://math.fontein.de/formulae/t6ueWtWakzOc_bRQuGKpq7NWCrXp9Boi3AQWsQ.svgz" alt="w \in W" title="w \in W"></span>, it follows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="14" src="https://math.fontein.de/formulae/juHwM8Eskl0sYO6Mst5Sk2lC7OnuRpAEQRAOtw.svgz" alt="\alpha_i = 0" title="\alpha_i = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. But then, <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="11" src="https://math.fontein.de/formulae/GDv2K9YNvA3j0.yxpqdxaQAdP78782LTKgUQdw.svgz" alt="x = 0" title="x = 0"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="12" src="https://math.fontein.de/formulae/is9hP3lsYTtjsAfjsj3Vw6gMuwWNAFVCFEhYNA.svgz" alt="\ker \Phi = 0" title="\ker \Phi = 0"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/h1YPUpOWbsd4f48ktllbLrRQbkfVpcJAOVaMmA.svgz" alt="\Phi" title="\Phi"></span> is injective.
</p>
<p>
Now let <span class="inline-formula"><img class="img-inline-formula img-formula" width="145" height="22" src="https://math.fontein.de/formulae/RLq1pRa7NyZ43l_SqUaQ2BhdLJfTQwgqPATPmQ.svgz" alt="\varphi \in \Hom_K^{fin}(W, V)" title="\varphi \in \Hom_K^{fin}(W, V)"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="18" src="https://math.fontein.de/formulae/zt1itL6D8gU1KsAconD3EjSKDrQySl4EWvudTQ.svgz" alt="(v_1, \dots, v_n)" title="(v_1, \dots, v_n)"></span> be a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="18" src="https://math.fontein.de/formulae/dHJXlxvkEBpFqx.eEpeqCSkNLbz51ialp89l2g.svgz" alt="\varphi(W)" title="\varphi(W)"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="18" src="https://math.fontein.de/formulae/Yrz5N.DVleAlyW2V_oKgWRD_w7mJXUWSeapyyg.svgz" alt="\pi_i : \varphi(W) \to K" title="\pi_i : \varphi(W) \to K"></span> be the projections with <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/ZZXx5bmU3AbVd8lF.octO87w5dNlFjYgYT9APQ.svgz" alt="\pi_i(v_i) = 1" title="\pi_i(v_i) = 1"></span> an <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/tioZcnv76JhKuHhjrA5gO4aaH0DfKzgh5cwP2Q.svgz" alt="\pi_i(v_j) = 0" title="\pi_i(v_j) = 0"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="38" height="16" src="https://math.fontein.de/formulae/sdNdxAE_xOEv38foNeA4HVOc.un.3BWONUcHgw.svgz" alt="i \neq j" title="i \neq j"></span>. Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="90" height="11" src="https://math.fontein.de/formulae/ULfEbT3kiGesKYuEezXo8noV0_fdd3lPIoskBg.svgz" alt="\alpha_i := \pi_i \circ \varphi" title="\alpha_i := \pi_i \circ \varphi"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="166" height="20" src="https://math.fontein.de/formulae/FpVaiSPH5TkcvHMcLlKB6.8Dq4KViNa8_hD5PA.svgz" alt="\varphi(w) = \sum_{i=1}^n \alpha_i(w) v_i" title="\varphi(w) = \sum_{i=1}^n \alpha_i(w) v_i"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="13" src="https://math.fontein.de/formulae/t6ueWtWakzOc_bRQuGKpq7NWCrXp9Boi3AQWsQ.svgz" alt="w \in W" title="w \in W"></span> since <span class="inline-formula"><img class="img-inline-formula img-formula" width="132" height="20" src="https://math.fontein.de/formulae/g8Hjvw4BQSGJlSWoF6gcQYNQcBxGn2O.4g2J9A.svgz" alt="v = \sum_{i=1}^n \pi_i(v) v_i" title="v = \sum_{i=1}^n \pi_i(v) v_i"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/VHMetKneqy.WdslRWXBdkeibydBQ5hNYNfFljw.svgz" alt="v \in \varphi(W)" title="v \in \varphi(W)"></span>; therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="159" height="20" src="https://math.fontein.de/formulae/NbiNYbpclh8QvU7ZffS0Qka7ur.ZrX36CXv2uQ.svgz" alt="\varphi = \Phi(\sum_{i=1}^n \alpha_i \tensor v_i)" title="\varphi = \Phi(\sum_{i=1}^n \alpha_i \tensor v_i)"></span>. This shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="237" height="22" src="https://math.fontein.de/formulae/OPGhyEo1OY5NM4CuOKwOk.ebzJUOtXS0WF6q2Q.svgz" alt="\Hom_K^{fin}(W, V) \subseteq \Phi(W^* \tensor_K V)" title="\Hom_K^{fin}(W, V) \subseteq \Phi(W^* \tensor_K V)"></span>, whence we have equality.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="22" src="https://math.fontein.de/formulae/OO_C61EurblJmqeN3RFHLX8fDc6nxDYo75DWpQ.svgz" alt="\Hom_K^{fin}(V, V)" title="\Hom_K^{fin}(V, V)"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebra, whence for <span class="inline-formula"><img class="img-inline-formula img-formula" width="160" height="22" src="https://math.fontein.de/formulae/rGscfF6ZchqBZDZnnY7rlNUE2Vn2JpD2w4WmrA.svgz" alt="\varphi, \psi \in \Hom_K^{fin}(V, V)" title="\varphi, \psi \in \Hom_K^{fin}(V, V)"></span>, it makes sense to define <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="16" src="https://math.fontein.de/formulae/JepEi3XBILTBpZeTRJAb53TebYjqEK_9PuZ6Ng.svgz" alt="\varphi \circ \psi : V \to V" title="\varphi \circ \psi : V \to V"></span>. We are interested on how <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="19" src="https://math.fontein.de/formulae/Nfq4aJ89xnewEVJwp_qewGvg_.VJ32r5WZOouw.svgz" alt="\Psi^{-1}(\varphi \circ \psi)" title="\Psi^{-1}(\varphi \circ \psi)"></span> can be described in terms of <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="19" src="https://math.fontein.de/formulae/0E4i0qIKPqjWMCJBlV5wY5a_j1WMWHE2VVNWCA.svgz" alt="\Psi^{-1}(\varphi)" title="\Psi^{-1}(\varphi)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="19" src="https://math.fontein.de/formulae/CVv84APyCKdxLcGZkOj5nUuiSqls0IxSxa50DA.svgz" alt="\Psi^{-1}(\psi)" title="\Psi^{-1}(\psi)"></span>. This is resolved by the following result:
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="16" src="https://math.fontein.de/formulae/V0tc7FPPZUXFRZWy7o.ijYdfmHOjGmf5snxO9A.svgz" alt="V, W, U" title="V, W, U"></span> be <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector spaces. The map
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="685" height="79" src="https://math.fontein.de/formulae/1UEi1hE_8P9zRVEUa7zXOc8d3VHL0hgfELTcHA.svgz" alt="m :{} & (W^* \tensor_K V) \times (U^* \tensor_K W) \to U^* \tensor_K V, \\
\biggl(\sum_{i=1}^n \beta_i \tensor v_i, \sum_{j=1}^m \alpha_j \tensor w_j\biggr) \mapsto \sum_{i=1}^n \sum_{j=1}^m \alpha_j \tensor \beta_i(w_j) v_i" title="m :{} & (W^* \tensor_K V) \times (U^* \tensor_K W) \to U^* \tensor_K V, \\
\biggl(\sum_{i=1}^n \beta_i \tensor v_i, \sum_{j=1}^m \alpha_j \tensor w_j\biggr) \mapsto \sum_{i=1}^n \sum_{j=1}^m \alpha_j \tensor \beta_i(w_j) v_i">
</div>
<p>
is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear and the following diagram commutes:
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="407" height="94" src="https://math.fontein.de/formulae/ahL8EpFt4t8MtccRNwSkHW3JMkadX12P24Y35w.svgz" alt="\xymatrix{ (W^* \tensor_K V) \times (U^* \tensor_K W) \ar[r]^{\qquad\quad m} \ar[d]^{\cong} & U^* \tensor_K V \ar[d]^{\cong} \\
\Hom_K^{fin}(W, V) \times \Hom_K^{fin}(U, W) \ar[r]_{\qquad\quad \circ} & \Hom_K^{fin}(U, V) }" title="\xymatrix{ (W^* \tensor_K V) \times (U^* \tensor_K W) \ar[r]^{\qquad\quad m} \ar[d]^{\cong} & U^* \tensor_K V \ar[d]^{\cong} \\
\Hom_K^{fin}(W, V) \times \Hom_K^{fin}(U, W) \ar[r]_{\qquad\quad \circ} & \Hom_K^{fin}(U, V) }">
</div>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="251" height="22" src="https://math.fontein.de/formulae/T8PLwRH4UxtfId57p4J7HgKBJpPSgB_5zM3OFA.svgz" alt="\Psi_1 : W^* \tensor_K V \to \Hom_K^{fin}(W, V)" title="\Psi_1 : W^* \tensor_K V \to \Hom_K^{fin}(W, V)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="251" height="22" src="https://math.fontein.de/formulae/MXm8OC3WYvDDNLmQxe6SF0YS6f3r5sJcImllBg.svgz" alt="\Psi_2 : U^* \tensor_K W \to \Hom_K^{fin}(U, W)" title="\Psi_2 : U^* \tensor_K W \to \Hom_K^{fin}(U, W)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="242" height="22" src="https://math.fontein.de/formulae/DsWEuxHLwd5c31d0BfH3.JxjeLUM6B7xA6Y6aA.svgz" alt="\Psi_3 : U^* \tensor_K V \to \Hom_K^{fin}(U, V)" title="\Psi_3 : U^* \tensor_K V \to \Hom_K^{fin}(U, V)"></span> be the canonical maps. Since these are isomorphisms, we have to show that for <span class="inline-formula"><img class="img-inline-formula img-formula" width="226" height="20" src="https://math.fontein.de/formulae/o.vAAOUu_bXVF2u0NCqWu6QcwKKSwySHQdeFfA.svgz" alt="x = \sum_{i=1}^n \beta_i \tensor v_i \in W^* \tensor_K V" title="x = \sum_{i=1}^n \beta_i \tensor v_i \in W^* \tensor_K V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="209" height="22" src="https://math.fontein.de/formulae/qs283Z6qqIXDzzXYJsc4Fmv77NxYayMrp0noEg.svgz" alt="y = \sum_{j=1}^m \alpha_j \tensor v_j U^* \tensor_K W" title="y = \sum_{j=1}^m \alpha_j \tensor v_j U^* \tensor_K W"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="319" height="22" src="https://math.fontein.de/formulae/Y1XcEAkoD8rALcbHxSuMURmoUN95DiFZgiSNdA.svgz" alt="z = \sum_{i=1}^n \sum_{j=1}^m \alpha_j \tensor \beta_i(w_j) v_i \in U^* \tensor_K V" title="z = \sum_{i=1}^n \sum_{j=1}^m \alpha_j \tensor \beta_i(w_j) v_i \in U^* \tensor_K V"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="175" height="18" src="https://math.fontein.de/formulae/5op_i9cqURY8KoVJjgW3lIPn.mbhSfRWfzpIhg.svgz" alt="\Psi_1(x) \circ \Psi_2(y) = \Psi_3(z)" title="\Psi_1(x) \circ \Psi_2(y) = \Psi_3(z)"></span>. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/pbepddnQV8SHGzjV9CeqUjvxje5osjQsHgcL5A.svgz" alt="u \in U" title="u \in U"></span>. Then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="478" height="174" src="https://math.fontein.de/formulae/bBABUhetw.ExDCWZDWf.xMGvhSjoiChDc6COLw.svgz" alt="(\Psi_1(x) \circ \Psi_2(y))(u) ={} & \Psi_1(x)(\Psi_2(y)(u)) = \Psi_1(x)\biggl( \sum_{j=1}^m \alpha_j(u) v_j \biggr) \\
{}={} & \sum_{i=1}^n \beta_i\biggl( \sum_{j=1}^m \alpha_j(u) v_j \biggr) v_i \\ {}={} & \sum_{i=1}^n \sum_{j=1}^m \alpha_j(u) \beta_i(v_j) v_i = \Psi_3(z)(u)," title="(\Psi_1(x) \circ \Psi_2(y))(u) ={} & \Psi_1(x)(\Psi_2(y)(u)) = \Psi_1(x)\biggl( \sum_{j=1}^m \alpha_j(u) v_j \biggr) \\
{}={} & \sum_{i=1}^n \beta_i\biggl( \sum_{j=1}^m \alpha_j(u) v_j \biggr) v_i \\ {}={} & \sum_{i=1}^n \sum_{j=1}^m \alpha_j(u) \beta_i(v_j) v_i = \Psi_3(z)(u),">
</div>
<p>
what we had to show.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="15" src="https://math.fontein.de/formulae/Vo0lUiqfySArrhRvi1aDeSi5467OPXvUWsXQJw.svgz" alt="V^* \tensor_K V" title="V^* \tensor_K V"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebra isomorphic to <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="22" src="https://math.fontein.de/formulae/OO_C61EurblJmqeN3RFHLX8fDc6nxDYo75DWpQ.svgz" alt="\Hom_K^{fin}(V, V)" title="\Hom_K^{fin}(V, V)"></span>; it posseses a <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/o.f5nkkZC1PEov1Bjz5CUC3EQ.MnIDCsxksltQ.svgz" alt="1" title="1"></span> if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="102" height="15" src="https://math.fontein.de/formulae/vqJkMFee.Cr2IwRQoTvDDMJn7oB5CYckXwjTQA.svgz" alt="\dim_K V < \infty" title="\dim_K V < \infty"></span>.
</p>
<p>
Now consider transposition
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="464" height="53" src="https://math.fontein.de/formulae/y4RVl.EaFuGtg4yW.yydNRdik0HI1TApvPdHlQ.svgz" alt="T : \Hom_K(V, W) \to \Hom_K(W^*, V^*), \quad \varphi \mapsto \begin{cases} W^* \to V^*, \\ \psi \mapsto \psi \circ \varphi. \end{cases}" title="T : \Hom_K(V, W) \to \Hom_K(W^*, V^*), \quad \varphi \mapsto \begin{cases} W^* \to V^*, \\ \psi \mapsto \psi \circ \varphi. \end{cases}">
</div>
<p>
Clearly, transposition is injective:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
The map <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear and injective.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
It is clear that <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear. To see that it is injective, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/ZMC.hxaJjVrSpZsWvlS99mLaGhPd3Z5vqZSb3Q.svgz" alt="\varphi \in \Hom_K(V, W)" title="\varphi \in \Hom_K(V, W)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="18" src="https://math.fontein.de/formulae/LKo.fadlwpUhbAt0GISKOO.QwTW53zaOduWQdw.svgz" alt="T(\varphi) = 0" title="T(\varphi) = 0"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="93" height="18" src="https://math.fontein.de/formulae/xjrCM6bKpc4FwTiesZizB8tJo00yeEJawUpGTw.svgz" alt="\psi(\varphi(v)) = 0" title="\psi(\varphi(v)) = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="16" src="https://math.fontein.de/formulae/AzfzcfrB9mppvzTLqDa8G.eXVgYRHfQK5gBB.Q.svgz" alt="\psi \in W^*" title="\psi \in W^*"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="18" src="https://math.fontein.de/formulae/D.g8nCeKGylCSH_m66TT5XyEgDZjdbBMYd9S8g.svgz" alt="\varphi(v) = 0" title="\varphi(v) = 0"></span> by <a href="https://math.fontein.de/2010/01/29/homomorphisms-tensor-products-and-certain-canonical-maps/#nonzeroform">the first lemma</a>. But that means <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="15" src="https://math.fontein.de/formulae/VIWb9NwIbMrRQMLcZvGvMMML8aJPFr_gtM1rmw.svgz" alt="\varphi = 0" title="\varphi = 0"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
We show that transposition restricts to the subspaces of the homomorphism spaces of homomorphisms with finite-dimensional image.
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/ZMC.hxaJjVrSpZsWvlS99mLaGhPd3Z5vqZSb3Q.svgz" alt="\varphi \in \Hom_K(V, W)" title="\varphi \in \Hom_K(V, W)"></span>. The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="277" height="18" src="https://math.fontein.de/formulae/lvbEh5I2ga594jbcheLwf9kCa7mXv_S9gwtwOg.svgz" alt="\varphi(V)^* \to T(\varphi)(W^*), \qquad \alpha \mapsto \alpha \circ \varphi" title="\varphi(V)^* \to T(\varphi)(W^*), \qquad \alpha \mapsto \alpha \circ \varphi">
</div>
<p>
is an isomorphism. In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="312" height="22" src="https://math.fontein.de/formulae/ZqyIfg9uohrp_LSLyel9YehjQpGHx.asOL7Zcg.svgz" alt="T^{-1}(\Hom_K^{fin}(W^*, V^*)) = \Hom_K^{fin}(V, W)" title="T^{-1}(\Hom_K^{fin}(W^*, V^*)) = \Hom_K^{fin}(V, W)"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/ZMC.hxaJjVrSpZsWvlS99mLaGhPd3Z5vqZSb3Q.svgz" alt="\varphi \in \Hom_K(V, W)" title="\varphi \in \Hom_K(V, W)"></span>. The map <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/MTVBtZ7lf9B1AJVlKfyBXtQADgzpgnBeHkjulA.svgz" alt="\psi : \varphi(V)^* \to T(\varphi)(W^*)" title="\psi : \varphi(V)^* \to T(\varphi)(W^*)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="11" src="https://math.fontein.de/formulae/XRZG8xvE_PFPtkQ6ExXOc5oC67uOiI1u.pc17Q.svgz" alt="\alpha \mapsto \alpha \circ \varphi" title="\alpha \mapsto \alpha \circ \varphi"></span> is well-defined and a homomorphism as <span class="inline-formula"><img class="img-inline-formula img-formula" width="125" height="18" src="https://math.fontein.de/formulae/0KlckONSFZkGDMEpILatvXORRZIVsxe8QuLbPg.svgz" alt="T(\varphi)(W^*) \subseteq V^*" title="T(\varphi)(W^*) \subseteq V^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="18" src="https://math.fontein.de/formulae/N9rEvTn2_KZrpU8RoCaK5K8IXDgyME1SzUZBFg.svgz" alt="\varphi(V) \subseteq W" title="\varphi(V) \subseteq W"></span>. Now let <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/Bh4jO.YwGp7asPko4GLQV2mNqF0n2qVSDmoOsg.svgz" alt="\alpha \in \varphi(V)^*" title="\alpha \in \varphi(V)^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="18" src="https://math.fontein.de/formulae/j0qUn8j90X.tjuqd4OykyQG8Kmxox1wgoGOqiw.svgz" alt="\psi(\alpha) = 0" title="\psi(\alpha) = 0"></span>, i.e. with <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="15" src="https://math.fontein.de/formulae/zCdtb2qJGEZwUs_UZ_hu8w0v1kX62nKcVqAe7Q.svgz" alt="\alpha \circ \varphi = 0" title="\alpha \circ \varphi = 0"></span>. But since <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/FomMDAW1Z5QiOIvnrijZY3Ehx1KX2HStvJR3Pg.svgz" alt="\alpha" title="\alpha"></span> is defined on <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/bvBRCXpN4bIAKW43JDNa6Hmo.SqKCjYFyhqMuA.svgz" alt="\varphi(V)" title="\varphi(V)"></span>, this means that <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="11" src="https://math.fontein.de/formulae/AfOJi9HmI9FSiqHpcwqh8DsHIf3DXCnVSbmDpg.svgz" alt="\alpha = 0" title="\alpha = 0"></span>. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="16" src="https://math.fontein.de/formulae/5YQhZpth_qEZfoZavf10MlpXOOEYGw_G8IF6hw.svgz" alt="\psi" title="\psi"></span> is injective.
</p>
<p>
Now let <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="18" src="https://math.fontein.de/formulae/BUGiZlZPX90VSe4kJxODXiI1bXYhPLek6KGl.A.svgz" alt="\beta \in T(\varphi)(W^*)" title="\beta \in T(\varphi)(W^*)"></span>, i.e. there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="20" src="https://math.fontein.de/formulae/wEwHyMM5pSBGFuOrpoeDGb4LwGTxwMAxvdHpJA.svgz" alt="\hat{\psi} \in W^*" title="\hat{\psi} \in W^*"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="20" src="https://math.fontein.de/formulae/1Bwt9U8DViTQ5bXPYQsBj5SLPWU3onE590No5A.svgz" alt="\beta = \hat{\psi} \circ \varphi" title="\beta = \hat{\psi} \circ \varphi"></span>. Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="90" height="23" src="https://math.fontein.de/formulae/CWCT8026E5UOVOF0HtDBeODt9dOlzhl1RUNXvg.svgz" alt="\alpha := \hat{\psi}|_{\varphi(V)}" title="\alpha := \hat{\psi}|_{\varphi(V)}"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/Bh4jO.YwGp7asPko4GLQV2mNqF0n2qVSDmoOsg.svgz" alt="\alpha \in \varphi(V)^*" title="\alpha \in \varphi(V)^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="238" height="23" src="https://math.fontein.de/formulae/uP.EM_3t7xdNmt_QOwIIdXvJDE1_xP3Xp50j4w.svgz" alt="\psi(\alpha) = \hat{\psi}|_{\varphi(V)} \circ \varphi = \hat{\psi} \circ \varphi = \beta" title="\psi(\alpha) = \hat{\psi}|_{\varphi(V)} \circ \varphi = \hat{\psi} \circ \varphi = \beta"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="16" src="https://math.fontein.de/formulae/5YQhZpth_qEZfoZavf10MlpXOOEYGw_G8IF6hw.svgz" alt="\psi" title="\psi"></span> is injective.
</p>
<p>
Finally, in case <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="18" src="https://math.fontein.de/formulae/JD1Hz6GaPJn2xqglN.QDzrR8FUfmY22VeyFPSQ.svgz" alt="\dim_K \varphi(V) < \infty" title="\dim_K \varphi(V) < \infty"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/WnJl4TZmbh2PpxQBfnikkgcpBKpADEKskJk_qA.svgz" alt="\dim_K \varphi(V)^* < \infty" title="\dim_K \varphi(V)^* < \infty"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="167" height="18" src="https://math.fontein.de/formulae/z8BNoJdNaXqE0.PMnnhgSB05lSxPhhE6XF2fPA.svgz" alt="\dim_K T(\varphi)(W^*) < \infty" title="\dim_K T(\varphi)(W^*) < \infty"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="191" height="22" src="https://math.fontein.de/formulae/N5Ofe5sUi4JlV3dK18_Xvcwm9iBfhf0.BIf8tw.svgz" alt="T(\varphi) \in \Hom_K^{fin}(W^*, V^*)" title="T(\varphi) \in \Hom_K^{fin}(W^*, V^*)"></span>. On the contrary, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="18" src="https://math.fontein.de/formulae/ZPP2pvpoaeHTMxLwxadV6tf7b1z7e448.LdMDg.svgz" alt="\dim_K \varphi(V) = \infty" title="\dim_K \varphi(V) = \infty"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="284" height="18" src="https://math.fontein.de/formulae/kVniWrMaq08F74IKn_9nAZppZcLDzwYxM78FlA.svgz" alt="\infty = \dim_K \varphi(V)^* = \dim_K T(\varphi)(W^*)" title="\infty = \dim_K \varphi(V)^* = \dim_K T(\varphi)(W^*)"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="191" height="22" src="https://math.fontein.de/formulae/pVQz60Z6eYvrXNzSZURZ1bddQV6GxZP7eyNQtw.svgz" alt="T(\varphi) \not\in \Hom_K^{fin}(W^*, V^*)" title="T(\varphi) \not\in \Hom_K^{fin}(W^*, V^*)"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now we have seen that <span class="inline-formula"><img class="img-inline-formula img-formula" width="211" height="22" src="https://math.fontein.de/formulae/IN1HDWxxYa7h17AbwDQArxafgOsrYAb4VGYihg.svgz" alt="\Hom_K^{fin}(V, W) \cong V^* \tensor_K W" title="\Hom_K^{fin}(V, W) \cong V^* \tensor_K W"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="245" height="22" src="https://math.fontein.de/formulae/jwN.GpLfr3yOllHGDGPXZG3AuiqCkVr_hL5Uwg.svgz" alt="\Hom_K^{fin}(W^*, V^*) \cong W^{**} \tensor_K V^*" title="\Hom_K^{fin}(W^*, V^*) \cong W^{**} \tensor_K V^*"></span> in a canonical way, and we have the canonical monomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="110" height="12" src="https://math.fontein.de/formulae/uRE900nytQyHxPdDyUTZQ4GC9SNVkyLApVtvkw.svgz" alt="\Psi : W \to W^{**}" title="\Psi : W \to W^{**}"></span>. We show that these maps behave nicely with transposition:
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
The map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="487" height="52" src="https://math.fontein.de/formulae/KZy9iP_wxHlMuHknm3vmSt_0rMbw6gLKyAz.dA.svgz" alt="T : V^* \tensor_K W \to W^{**} \tensor_K V^*, \qquad \sum_{i=1}^n v_i^* \tensor w_i \mapsto \sum_{i=1}^n \Psi(w_i) \tensor v_i^*" title="T : V^* \tensor_K W \to W^{**} \tensor_K V^*, \qquad \sum_{i=1}^n v_i^* \tensor w_i \mapsto \sum_{i=1}^n \Psi(w_i) \tensor v_i^*">
</div>
<p>
is the unique homomorphism which makes the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="307" height="110" src="https://math.fontein.de/formulae/7oMSbmu2Bqvg0EJia1ci6YSPGyt3G2nnBiiTGw.svgz" alt="\xymatrix{ \Hom_K^{fin}(V, W) \ar[r]^T \ar[d]_{\cong} & \Hom_K^{fin}(W^*, V^*) \ar[d]^{\cong} \\
V^* \tensor_K W \ar[r]_T & W^{**} \tensor_K V^* }" title="\xymatrix{ \Hom_K^{fin}(V, W) \ar[r]^T \ar[d]_{\cong} & \Hom_K^{fin}(W^*, V^*) \ar[d]^{\cong} \\
V^* \tensor_K W \ar[r]_T & W^{**} \tensor_K V^* }">
</div>
<p>
commuting.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="231" height="20" src="https://math.fontein.de/formulae/.uzQakKtn88BCR0AQDnhGK54rQOqdiaw8Szhyg.svgz" alt="x = \sum_{i=1}^n v_i^* \tensor w_i \in V^* \tensor_K W" title="x = \sum_{i=1}^n v_i^* \tensor w_i \in V^* \tensor_K W"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="274" height="20" src="https://math.fontein.de/formulae/Odm5Ma6gdVUxTONEr3u6_XZ1RgsBlsa5_H.cSw.svgz" alt="y = \sum_{i=1}^n \Psi(w_i) \tensor v_i^* \in W^{**} \tensor_K V^*" title="y = \sum_{i=1}^n \Psi(w_i) \tensor v_i^* \in W^{**} \tensor_K V^*"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="229" height="20" src="https://math.fontein.de/formulae/4l1jvFqGhgl_3mySQW4zA.44kZeHZkLaPt8yFQ.svgz" alt="\Phi(x)(v) = \sum_{i=1}^n v_i^*(v) w_i \in W" title="\Phi(x)(v) = \sum_{i=1}^n v_i^*(v) w_i \in W"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, and
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="440" height="223" src="https://math.fontein.de/formulae/bx3GybEfHyxYyZ_vvOOl.k.RbLdPfUjc_gEiCg.svgz" alt="& T(\Phi(x))(w^*)(v) = (w^* \circ \Phi(x))(v) = w^*(\Phi(x)(v)) \\
{}={} & w^*\biggl(\sum_{i=1}^n v_i^*(v) w_i\biggr) = \sum_{i=1}^n v_i^*(v) w^*(w_i) \\
{}={} & \sum_{i=1}^n v_i^*(v) \Psi(w_i)(w^*) = \biggl( \sum_{i=1}^n w^*(w_i) v_i^* \biggr)(v) \\
{}={} & \biggl( \sum_{i=1}^n \Psi(w_i)(w^*) v_i^* \biggr)(v) = \biggl( \sum_{i=1}^n \Phi(\Psi(w_i) \tensor v_i^*)(w^*) \biggr)(v) \\
{}={} & \Phi(y)(w^*)(v)" title="& T(\Phi(x))(w^*)(v) = (w^* \circ \Phi(x))(v) = w^*(\Phi(x)(v)) \\
{}={} & w^*\biggl(\sum_{i=1}^n v_i^*(v) w_i\biggr) = \sum_{i=1}^n v_i^*(v) w^*(w_i) \\
{}={} & \sum_{i=1}^n v_i^*(v) \Psi(w_i)(w^*) = \biggl( \sum_{i=1}^n w^*(w_i) v_i^* \biggr)(v) \\
{}={} & \biggl( \sum_{i=1}^n \Psi(w_i)(w^*) v_i^* \biggr)(v) = \biggl( \sum_{i=1}^n \Phi(\Psi(w_i) \tensor v_i^*)(w^*) \biggr)(v) \\
{}={} & \Phi(y)(w^*)(v)">
</div>
<p>
for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="13" src="https://math.fontein.de/formulae/s1st0.qawKFkGGaV2eHradAby9F8qnCroWiecQ.svgz" alt="w^* \in W^*" title="w^* \in W^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="18" src="https://math.fontein.de/formulae/.QMPtj7G1zZa5vZ7fDtYW10c17OPdHnY38If6w.svgz" alt="T(\Phi(x)) = \Phi(y)" title="T(\Phi(x)) = \Phi(y)"></span>, what we had to show.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now consider double transposition, i.e.
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="329" height="18" src="https://math.fontein.de/formulae/LigdxSSfTHPblJ6FlcK0ihmc1hFB7wEcwZlTUg.svgz" alt="T \circ T : \Hom_K(V, W) \to \Hom_K(V^{**}, W^{**})," title="T \circ T : \Hom_K(V, W) \to \Hom_K(V^{**}, W^{**}),">
</div>
<p>
and its finite-dimensional image restriction
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="347" height="22" src="https://math.fontein.de/formulae/.c1TEoquhQQwijRz9P7lgxCyw7A5.DYhv_UAgw.svgz" alt="T \circ T : \Hom_K^{fin}(V, W) \to \Hom_K^{fin}(V^{**}, W^{**})." title="T \circ T : \Hom_K^{fin}(V, W) \to \Hom_K^{fin}(V^{**}, W^{**}).">
</div>
<p>
The above shows that using the canonical isomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="211" height="22" src="https://math.fontein.de/formulae/IN1HDWxxYa7h17AbwDQArxafgOsrYAb4VGYihg.svgz" alt="\Hom_K^{fin}(V, W) \cong V^* \tensor_K W" title="\Hom_K^{fin}(V, W) \cong V^* \tensor_K W"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="274" height="22" src="https://math.fontein.de/formulae/p1CMgNkG6lgEFXXgLPOndPWkdMQsOZazsfbKoQ.svgz" alt="\Hom_K^{fin}(V^{**}, W^{**}) \cong V^{***} \tensor_K W^{**}" title="\Hom_K^{fin}(V^{**}, W^{**}) \cong V^{***} \tensor_K W^{**}"></span>, we can describe double transpotition by the following commuting diagram:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="332" height="148" src="https://math.fontein.de/formulae/MZdD1DT3ioXB0ym0HHdzW4J.DgxN0NoqGz7yQg.svgz" alt="\xymatrix@R-0.85cm{ \Hom_K^{fin}(V, W) \ar[r]^{T \circ T \;\;} \ar[dddd]_{\cong} & \Hom_K^{fin}(V^{**}, W^{**}) \ar[dddd]^{\cong} \\
{\vphantom{x}} \\ {\vphantom{y}} \\ {\vphantom{z}} \\ V^* \tensor_K W \ar[r]^{T \circ T \;\;} & V^{***} \tensor_K W^{**} \\
\sum_{i=1}^n v_i^* \tensor w_i \ar@{|->}[r] & \sum_{i=1}^n \Psi(v_i^*) \tensor \Psi(w_i) }" title="\xymatrix@R-0.85cm{ \Hom_K^{fin}(V, W) \ar[r]^{T \circ T \;\;} \ar[dddd]_{\cong} & \Hom_K^{fin}(V^{**}, W^{**}) \ar[dddd]^{\cong} \\
{\vphantom{x}} \\ {\vphantom{y}} \\ {\vphantom{z}} \\ V^* \tensor_K W \ar[r]^{T \circ T \;\;} & V^{***} \tensor_K W^{**} \\
\sum_{i=1}^n v_i^* \tensor w_i \ar@{|->}[r] & \sum_{i=1}^n \Psi(v_i^*) \tensor \Psi(w_i) }">
</div>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="18" src="https://math.fontein.de/formulae/k97uHOgUpJbDgqLdmlAdsK2p0U2YnpuJdpC6eg.svgz" alt="\psi \in \Hom_K(W^*, V^*)" title="\psi \in \Hom_K(W^*, V^*)"></span>, we obtain a map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="352" height="53" src="https://math.fontein.de/formulae/wKj9u57A9sHIBdxwcZnVo4VTVKZIKPgF9DBHzg.svgz" alt="H(\psi) : V \to W^{**}, \qquad v \mapsto \begin{cases} W^* \to K \\ \alpha \mapsto \psi(\alpha)(v). \end{cases}" title="H(\psi) : V \to W^{**}, \qquad v \mapsto \begin{cases} W^* \to K \\ \alpha \mapsto \psi(\alpha)(v). \end{cases}">
</div>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
The map <span class="inline-formula"><img class="img-inline-formula img-formula" width="299" height="18" src="https://math.fontein.de/formulae/WKVGch_PoLdx8bicc5fK30Ppgo2zT1fyWV4Tww.svgz" alt="H : \Hom_K(W^*, V^*) \to \Hom_K(V, W^{**})" title="H : \Hom_K(W^*, V^*) \to \Hom_K(V, W^{**})"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear and injective.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
First, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="18" src="https://math.fontein.de/formulae/k97uHOgUpJbDgqLdmlAdsK2p0U2YnpuJdpC6eg.svgz" alt="\psi \in \Hom_K(W^*, V^*)" title="\psi \in \Hom_K(W^*, V^*)"></span> is fixed, <span class="inline-formula"><img class="img-inline-formula img-formula" width="302" height="18" src="https://math.fontein.de/formulae/8MkebCjikWGbZUrSttE5cl6Ni0YE2IQ0Ht4WGQ.svgz" alt="H(\psi)(v + \lambda v') = H(\psi)(v) + \lambda H(\psi)(v')" title="H(\psi)(v + \lambda v') = H(\psi)(v) + \lambda H(\psi)(v')"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="17" src="https://math.fontein.de/formulae/88G2lC9cJmBg5YnTtD1MgDuGjI7Z3XWTqIEcLw.svgz" alt="v, v' \in V" title="v, v' \in V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/4GBul8lr4xhjZ8qO606B_Uq61duCBNzLS5bGFQ.svgz" alt="\lambda \in K" title="\lambda \in K"></span>; hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="102" height="18" src="https://math.fontein.de/formulae/BmUwgMSZ08FziRTOr95aAa7pc432g_l6PZt8YA.svgz" alt="H(V) \subseteq W^{**}" title="H(V) \subseteq W^{**}"></span>. Now, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="181" height="18" src="https://math.fontein.de/formulae/9xh0vDuygnR3js_BV9NHzjqHgCqB07OjatOqtQ.svgz" alt="\psi, \psi' \in \Hom_K(W^*, V^*)" title="\psi, \psi' \in \Hom_K(W^*, V^*)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/4GBul8lr4xhjZ8qO606B_Uq61duCBNzLS5bGFQ.svgz" alt="\lambda \in K" title="\lambda \in K"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, we have
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="465" height="45" src="https://math.fontein.de/formulae/6qlDveE4c6AuCxg3HBZIsajIJ2ylZMcGKzXkPQ.svgz" alt="H(\psi + \lambda \psi')(v) ={} & (\psi + \lambda \psi')(\alpha)(v) = \alpha((\psi + \lambda \psi')(v)) \\
{}={} & \alpha(\psi(v) + \lambda \psi'(v)) = H(\psi)(v) + \lambda H(\psi)(v')," title="H(\psi + \lambda \psi')(v) ={} & (\psi + \lambda \psi')(\alpha)(v) = \alpha((\psi + \lambda \psi')(v)) \\
{}={} & \alpha(\psi(v) + \lambda \psi'(v)) = H(\psi)(v) + \lambda H(\psi)(v'),">
</div>
<p>
whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear.
</p>
<p>
To see that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is injective, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="18" src="https://math.fontein.de/formulae/k97uHOgUpJbDgqLdmlAdsK2p0U2YnpuJdpC6eg.svgz" alt="\psi \in \Hom_K(W^*, V^*)" title="\psi \in \Hom_K(W^*, V^*)"></span> be such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/hQMfRtEH4wexBU8RTGX1YG41FPTvjTmuMw3l7g.svgz" alt="H(\psi) = 0" title="H(\psi) = 0"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="13" src="https://math.fontein.de/formulae/tWzMm5xXdoo8fbQ48wx9erLefd2NOxQrcoYjVw.svgz" alt="\alpha \in W^*" title="\alpha \in W^*"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>; since <span class="inline-formula"><img class="img-inline-formula img-formula" width="93" height="18" src="https://math.fontein.de/formulae/n7dHr.LQxHQKlloQOREMh2Nlwin6gTBsX8cpWQ.svgz" alt="\psi(\alpha)(v) = 0" title="\psi(\alpha)(v) = 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/KsOArZbMnsJRrzwj8HWDL_L45lvecz0OO6uAFA.svgz" alt="v" title="v"></span>, we see that <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="18" src="https://math.fontein.de/formulae/j0qUn8j90X.tjuqd4OykyQG8Kmxox1wgoGOqiw.svgz" alt="\psi(\alpha) = 0" title="\psi(\alpha) = 0"></span>, but since this is the case for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/FomMDAW1Z5QiOIvnrijZY3Ehx1KX2HStvJR3Pg.svgz" alt="\alpha" title="\alpha"></span> we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="16" src="https://math.fontein.de/formulae/qpNNEbeOHec0LvchiEKbYg_W2E2wZNU1kn_uaQ.svgz" alt="\psi = 0" title="\psi = 0"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Note that we have the following diagram:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="493" height="170" src="https://math.fontein.de/formulae/SCyC0drIv27EOdjJ4zlFiLeVVOEn0IUfJgo6XQ.svgz" alt="\xymatrix{ & & \Hom_K(V, W) \ar[dl]_T \\ & \Hom_K(W^*, V^*) \ar[dl]_T \ar[dr]^H & \\ \Hom_K(V^{**}, W^{**}) & & \Hom_K(V, W^{**}) }" title="\xymatrix{ & & \Hom_K(V, W) \ar[dl]_T \\ & \Hom_K(W^*, V^*) \ar[dl]_T \ar[dr]^H & \\ \Hom_K(V^{**}, W^{**}) & & \Hom_K(V, W^{**}) }">
</div>
<p>
Moreover, using the canonical embeddings <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="12" src="https://math.fontein.de/formulae/S.5reg1mC6yDupvYuUQf3m2oWl8kBCuZzuXXyQ.svgz" alt="\Psi : V \to V^{**}" title="\Psi : V \to V^{**}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="110" height="12" src="https://math.fontein.de/formulae/uRE900nytQyHxPdDyUTZQ4GC9SNVkyLApVtvkw.svgz" alt="\Psi : W \to W^{**}" title="\Psi : W \to W^{**}"></span>, we can define a map <span class="inline-formula"><img class="img-inline-formula img-formula" width="282" height="18" src="https://math.fontein.de/formulae/J53tw2Ux8fQ8u21hBzj6GvuBd6HGycFY.j9YEg.svgz" alt="\Hom_K(V^{**}, W^{**}) \to \Hom_K(V, W^{**})" title="\Hom_K(V^{**}, W^{**}) \to \Hom_K(V, W^{**})"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/osZjYqPiKnSs5EyRQD4uO2VJfeENdCcJjDF8Ow.svgz" alt="\varphi \mapsto \varphi \circ \Phi" title="\varphi \mapsto \varphi \circ \Phi"></span>, and a map <span class="inline-formula"><img class="img-inline-formula img-formula" width="248" height="18" src="https://math.fontein.de/formulae/fZ2SYzoNq_CiRAJcYVBcCbd_weJ80SaWU5F_MQ.svgz" alt="\Hom_K(V, W) \to \Hom_K(V, W^{**})" title="\Hom_K(V, W) \to \Hom_K(V, W^{**})"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/ib.rSBiljXmJOlwfaUv.nHTSzUbCA9n_MepRzA.svgz" alt="\varphi \mapsto \Phi \circ \varphi" title="\varphi \mapsto \Phi \circ \varphi"></span>. It turns out that these map make the diagram commute:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
The maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="313" height="21" src="https://math.fontein.de/formulae/CbaKU_ynBrhmGKq2YBuaUP.9QUCf7wYFwenKcg.svgz" alt="\hat{H} : \Hom_K(V^{**}, W^{**}) \to \Hom_K(V, W^{**})" title="\hat{H} : \Hom_K(V^{**}, W^{**}) \to \Hom_K(V, W^{**})"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/osZjYqPiKnSs5EyRQD4uO2VJfeENdCcJjDF8Ow.svgz" alt="\varphi \mapsto \varphi \circ \Phi" title="\varphi \mapsto \varphi \circ \Phi"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="279" height="21" src="https://math.fontein.de/formulae/egHnxlKbnaj6WTKTRfUEo7jkPaqOUgRvr7dd7Q.svgz" alt="\tilde{H} : \Hom_K(V, W) \to \Hom_K(V, W^{**})" title="\tilde{H} : \Hom_K(V, W) \to \Hom_K(V, W^{**})"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/ib.rSBiljXmJOlwfaUv.nHTSzUbCA9n_MepRzA.svgz" alt="\varphi \mapsto \Phi \circ \varphi" title="\varphi \mapsto \Phi \circ \varphi"></span> are <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear and make the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="493" height="178" src="https://math.fontein.de/formulae/D0BwkiqO.eSGukBox4OLbei_ya0du1DcU1_62w.svgz" alt="\xymatrix{ & & \Hom_K(V, W) \ar[dl]_T \ar[dd]^{\tilde{H}} \\ & \Hom_K(W^*, V^*) \ar[dl]_T \ar[dr]^H & \\
\Hom_K(V^{**}, W^{**}) \ar[rr]_{\hat{H}} & & \Hom_K(V, W^{**}) }" title="\xymatrix{ & & \Hom_K(V, W) \ar[dl]_T \ar[dd]^{\tilde{H}} \\ & \Hom_K(W^*, V^*) \ar[dl]_T \ar[dr]^H & \\
\Hom_K(V^{**}, W^{**}) \ar[rr]_{\hat{H}} & & \Hom_K(V, W^{**}) }">
</div>
<p>
commute. In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/fdOUDnBBx7OMssbntUfkZw8mVivwzNpxYrsf8Q.svgz" alt="\tilde{H}" title="\tilde{H}"></span> is injective.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
That <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="17" src="https://math.fontein.de/formulae/kNi_3rNlr_oSpTFwklzH6AU4aYBeiVsQpvw0Ow.svgz" alt="\hat{H}" title="\hat{H}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/fdOUDnBBx7OMssbntUfkZw8mVivwzNpxYrsf8Q.svgz" alt="\tilde{H}" title="\tilde{H}"></span> are <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear is clear. For the lower triangle, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="155" height="18" src="https://math.fontein.de/formulae/Xdv0dsePlWiHBE2EGoM8lPE.rXlV2PMj70LWZw.svgz" alt="\varphi \in \Hom_K(W^*, V^*)" title="\varphi \in \Hom_K(W^*, V^*)"></span>; we have to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="21" src="https://math.fontein.de/formulae/OTWLwqmIAFpaLlQT4NwvNh57otZvqDrfID84ww.svgz" alt="\hat{H}(T(\varphi)) = H(\varphi)" title="\hat{H}(T(\varphi)) = H(\varphi)"></span>. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="13" src="https://math.fontein.de/formulae/tWzMm5xXdoo8fbQ48wx9erLefd2NOxQrcoYjVw.svgz" alt="\alpha \in W^*" title="\alpha \in W^*"></span>; then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="409" height="48" src="https://math.fontein.de/formulae/ag2tDLzpImpMYpqJ8fP6Lfk3Rpu28ELIHAdALQ.svgz" alt="H(\varphi)(v)(\alpha) ={} & \varphi(\alpha)(v) = \Phi(v)(\varphi(\alpha)) = (\Phi(v) \circ \varphi)(\alpha) \\
{}={} & T(\varphi)(\Phi(v))(\alpha) = \hat{H}(T(\varphi))(v)(\alpha)." title="H(\varphi)(v)(\alpha) ={} & \varphi(\alpha)(v) = \Phi(v)(\varphi(\alpha)) = (\Phi(v) \circ \varphi)(\alpha) \\
{}={} & T(\varphi)(\Phi(v))(\alpha) = \hat{H}(T(\varphi))(v)(\alpha).">
</div>
<p>
For the right triangle, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/ZMC.hxaJjVrSpZsWvlS99mLaGhPd3Z5vqZSb3Q.svgz" alt="\varphi \in \Hom_K(V, W)" title="\varphi \in \Hom_K(V, W)"></span>; we have to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="21" src="https://math.fontein.de/formulae/Vz1PViTDbKHEok6FTUeP3zvtVRCzZTJL8Ib8kQ.svgz" alt="H(T(\varphi)) = \tilde{H}(\varphi)" title="H(T(\varphi)) = \tilde{H}(\varphi)"></span>. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="13" src="https://math.fontein.de/formulae/tWzMm5xXdoo8fbQ48wx9erLefd2NOxQrcoYjVw.svgz" alt="\alpha \in W^*" title="\alpha \in W^*"></span>; then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="470" height="47" src="https://math.fontein.de/formulae/FAKCkyyzQnQOK_8LLCPQ_6GSq0qkiSLeJm82Dw.svgz" alt="H(T(\varphi))(v)(\alpha) ={} & T(\varphi)(\alpha)(v) = (\alpha \circ \varphi)(v) = \alpha(\varphi(v)) \\
{}={} & \Phi(\varphi(v))(\alpha) = (\Phi \circ \varphi)(v)(\alpha) = \tilde{H}(\varphi)(v)(\alpha)." title="H(T(\varphi))(v)(\alpha) ={} & T(\varphi)(\alpha)(v) = (\alpha \circ \varphi)(v) = \alpha(\varphi(v)) \\
{}={} & \Phi(\varphi(v))(\alpha) = (\Phi \circ \varphi)(v)(\alpha) = \tilde{H}(\varphi)(v)(\alpha).">
</div>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is injective. We can use this to determine the image of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span>. For example, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="156" height="18" src="https://math.fontein.de/formulae/tVENvpVib0Ge8lP68ImQmitHmG23SKEdDAEnDw.svgz" alt="\psi \in \Hom_K(V^*, W^*)" title="\psi \in \Hom_K(V^*, W^*)"></span>,
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="435" height="124" src="https://math.fontein.de/formulae/ymIfrPsPsyVXSUZ1QfJoW8gSuM8hTKpyP7Skqg.svgz" alt="& \exists \varphi \in \Hom_K(V, W) : T(\varphi) = \psi \\
{}\Leftrightarrow{} & \forall v \in V : H(\psi)(v) \in \Phi(W) \\
{}\Leftrightarrow{} & \forall v \in V : (\alpha \mapsto \psi(\alpha)(v)) \in \Phi(W) \\
{}\Leftrightarrow{} & \forall v \in V : \bigcap_{\alpha \in V^* : \psi(\alpha)(v) = 0} \ker \alpha = 0 \text{ implies } \psi(\bullet)(v) = 0;" title="& \exists \varphi \in \Hom_K(V, W) : T(\varphi) = \psi \\
{}\Leftrightarrow{} & \forall v \in V : H(\psi)(v) \in \Phi(W) \\
{}\Leftrightarrow{} & \forall v \in V : (\alpha \mapsto \psi(\alpha)(v)) \in \Phi(W) \\
{}\Leftrightarrow{} & \forall v \in V : \bigcap_{\alpha \in V^* : \psi(\alpha)(v) = 0} \ker \alpha = 0 \text{ implies } \psi(\bullet)(v) = 0;">
</div>
<p>
the last equivalence follows from the <a href="https://math.fontein.de/2010/01/29/homomorphisms-tensor-products-and-certain-canonical-maps/#Psimapprop">first proposition</a>. Unfortunately, this criterion does not really helps in practice.
</p>
<p>
In case anyone knows a better description of the image of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> or <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/gMWgy0TNdy9YCjmc.NeT.40720Hpf5GzYeko1A.svgz" alt="\Psi" title="\Psi"></span>, I'd be happy to know.
</p>
</div>
About Base Changes and Tensor Products.
https://math.fontein.de/2009/08/15/about-base-changes-and-tensor-products/
2009-08-15T19:48:25+02:00
2009-08-15T19:48:25+02:00
Felix Fontein
<div>
<p>
In introductionary Linear Algebra classes, one often has the following problems: let <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="15" src="https://math.fontein.de/formulae/0anVpxnNe.qZ9z.jSb51iyHOfUNSrlJP0BlCfQ.svgz" alt="A \in \R^{n \times n}" title="A \in \R^{n \times n}"></span> be a real valued matrix, say an orthogonal one, then the eigenvalues are complex numbers of absolute value 1. the only two such values inside <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span> are <span class="inline-formula"><img class="img-inline-formula img-formula" width="23" height="13" src="https://math.fontein.de/formulae/wZ9MAmJo6nwUSTgdTx0ziZPwpp6.JX9w8_HQjg.svgz" alt="\pm 1" title="\pm 1"></span>; hence, most eigenvalues of orthogonal matrices are not elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>. Now, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/IQHNYFOk0rOuhD6lWdQfJQcLyezSNAov1CulYw.svgz" alt="(V, \ggen{\bullet, \bullet})" title="(V, \ggen{\bullet, \bullet})"></span> be a finite-dimensional Euclidean space and <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="16" src="https://math.fontein.de/formulae/toKlST5l9Zq8DiYbJjeQyhVVYsuJ9Bml7WeO9A.svgz" alt="\phi : V \to V" title="\phi : V \to V"></span> an orthogonal map. If one fixes an orthogonal basis <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/Ih7V6Mp4twAjwq7Tr86Dt9U5c5VQJd0XWRPlqw.svgz" alt="B" title="B"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>, one obtains a orthogonal matrix <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/xH7tLVLa97ISeVHWGBEztxF8GLFrn5yyCJGpJg.svgz" alt="A = M_B(\phi)" title="A = M_B(\phi)"></span> which represents <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span>. One can talk about complex eigenvalues of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span>, but what about complex eigenvalues of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span>? What should these be? <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="12" src="https://math.fontein.de/formulae/8nUBhurVBj68zhlweME21wMzNdX0n_qrLHJrSA.svgz" alt="\lambda v" title="\lambda v"></span> does not make sense for a complex number <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/sO3R8KebAmzbuFRx.G.Si6tH7Mjh16B5xehK.Q.svgz" alt="\lambda" title="\lambda"></span>, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> is a vector space over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>.
</p>
<p>
The usual solution to this is to complexify <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>: define <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="15" src="https://math.fontein.de/formulae/39cYyOY5sV.nsT2YPG9R9ylROYJUabYpx2q2Rg.svgz" alt="V_\C := V \oplus V" title="V_\C := V \oplus V"></span>, and define an action
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="431" height="43" src="https://math.fontein.de/formulae/fpTWLrqx9x6_sEPze9Ei8XUQzVxWfPfQbJwDQQ.svgz" alt="& \C \times V_\C \to V_\C, \\
& (a + i b, (v, w)) \mapsto (a + i b) (v + i w) = (a v - b w, b v + a w);" title="& \C \times V_\C \to V_\C, \\
& (a + i b, (v, w)) \mapsto (a + i b) (v + i w) = (a v - b w, b v + a w);">
</div>
<p>
this turns <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span> into a <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>-vector space. If one identifies <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> by its image under <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="15" src="https://math.fontein.de/formulae/tj2BcMoOCmAX3l7pgcD1e_Hgnf8PbnlRabN.8g.svgz" alt="V \to V_\C" title="V \to V_\C"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="18" src="https://math.fontein.de/formulae/a3i9zxX62QwcwSHR2IAUxNWN3NMQgh.sOFGGPw.svgz" alt="v \mapsto (v, 0)" title="v \mapsto (v, 0)"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="144" height="18" src="https://math.fontein.de/formulae/LHcYwjCBfaiWrEXMnirv7FgCEZHg35sGdmMgxA.svgz" alt="\lambda v = (\lambda + 0 i) (v, 0)" title="\lambda v = (\lambda + 0 i) (v, 0)"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/f80IHiNCXPxkVeob5JEcg2jiEx.E2Cl2zsmt9A.svgz" alt="\lambda \in \R" title="\lambda \in \R"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>. Now we are left to extend <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span>. It turns out that there is exactly one choice to extend <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span> to a <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>-linear map <span class="inline-formula"><img class="img-inline-formula img-formula" width="103" height="16" src="https://math.fontein.de/formulae/U7oJMOLAWoXGx3P4U90F19M62PzaRkJPNwhSqw.svgz" alt="\phi_\C : V_\C \to V_\C" title="\phi_\C : V_\C \to V_\C"></span>, i.e. that <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/KBeJU.aKgJtravDyB7C5oPynQlv.SVniIE8CLQ.svgz" alt="\phi_\C|_V = \phi" title="\phi_\C|_V = \phi"></span>. Namely, one has to define <span class="inline-formula"><img class="img-inline-formula img-formula" width="186" height="18" src="https://math.fontein.de/formulae/6YAZrwhP_LBq3QP9OuZh._xn0TzEx57gtCVWXg.svgz" alt="\phi_\C(v, w) := (\phi(v), \phi(w))" title="\phi_\C(v, w) := (\phi(v), \phi(w))"></span>; this is obviously <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-linear, whence it suffices to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="178" height="18" src="https://math.fontein.de/formulae/TfTv1RfbfODN8jf9XHouc.L1_DPort3feki36Q.svgz" alt="\phi_\C(i (v, w)) = i \phi_\C(v, w)" title="\phi_\C(i (v, w)) = i \phi_\C(v, w)"></span>:
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="445" height="44" src="https://math.fontein.de/formulae/QhcMyPtSbar7zw5sCVeL4dAy4qUYntsjuobd_Q.svgz" alt="\phi_\C(i (v, w)) ={} & \phi_\C(-w, v) = (\phi(-w), \phi(v)) = (-\phi(w), \phi(v)) \\
{}={} & i (\phi(v), \phi(w)) = i \phi_\C(v, w)." title="\phi_\C(i (v, w)) ={} & \phi_\C(-w, v) = (\phi(-w), \phi(v)) = (-\phi(w), \phi(v)) \\
{}={} & i (\phi(v), \phi(w)) = i \phi_\C(v, w).">
</div>
<p>
Now if <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/Ih7V6Mp4twAjwq7Tr86Dt9U5c5VQJd0XWRPlqw.svgz" alt="B" title="B"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>, it is as well an <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span>; moreover, <span class="inline-formula"><img class="img-inline-formula img-formula" width="141" height="18" src="https://math.fontein.de/formulae/IO4BoPI.QaO1b9QZSXJmYqRW.YdG8yJNlosgLA.svgz" alt="M_B(\phi) = M_B(\phi_\C)" title="M_B(\phi) = M_B(\phi_\C)"></span>. If now <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/jfypCeE7ybMPnhD.TcSCALIZRLBZckuutt6fcw.svgz" alt="\lambda \in \C" title="\lambda \in \C"></span> is a complex eigenvalue of <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/sXRDv_yxGLPckx5.IWVoAnOO42UY9R14KcbPKw.svgz" alt="M_B(\phi)" title="M_B(\phi)"></span>, then there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="18" src="https://math.fontein.de/formulae/piFKNPVnLzSeESOB9CMVslUXoUI7I_2xsgP4KQ.svgz" alt="\hat{v} \in V_\C \setminus \{ 0 \}" title="\hat{v} \in V_\C \setminus \{ 0 \}"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="18" src="https://math.fontein.de/formulae/HK6z5vQk9APSk02iRrHubTv7xHXtfx8S.gaZgw.svgz" alt="\phi_\C(\hat{v}) = \lambda \hat{v}" title="\phi_\C(\hat{v}) = \lambda \hat{v}"></span>. So <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/sO3R8KebAmzbuFRx.G.Si6tH7Mjh16B5xehK.Q.svgz" alt="\lambda" title="\lambda"></span> is indeed an eigenvalue of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="16" src="https://math.fontein.de/formulae/9MW4KWJWz6ec6w6aUgZuggSn7Qc7C17smbsW6w.svgz" alt="\phi_\C" title="\phi_\C"></span>. Abusing notation, we say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/sO3R8KebAmzbuFRx.G.Si6tH7Mjh16B5xehK.Q.svgz" alt="\lambda" title="\lambda"></span> is an eigenvalue of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span>; this will always mean that we are talking of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="16" src="https://math.fontein.de/formulae/9MW4KWJWz6ec6w6aUgZuggSn7Qc7C17smbsW6w.svgz" alt="\phi_\C" title="\phi_\C"></span>. This process is called <em>complexification</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span>.
</p>
<p>
But does this generalize? What if <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="15" src="https://math.fontein.de/formulae/a1rDqZj45RVF5GbvFM5L5YIdy0CViFjph6A.wQ.svgz" alt="K = \F_2" title="K = \F_2"></span> is the base field and one has an eigenvalue <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="15" src="https://math.fontein.de/formulae/.YbfCMx_POcUQEeWWLyGAURMwOki4mfsHkONgg.svgz" alt="\lambda \in L = \F_8" title="\lambda \in L = \F_8"></span> of the matrix? Can we do the same thing here? And what if <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="15" src="https://math.fontein.de/formulae/UxlQfHmMNpMZSmsw4V0Yyc1xcO8HzcFZaHEI7g.svgz" alt="K = \Q" title="K = \Q"></span> and we have an eigenvalue in <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="12" src="https://math.fontein.de/formulae/tdWnrYR0uO8hdTrPgen62JDFpJhc30uAGQkrKw.svgz" alt="L = \C" title="L = \C"></span>? The answer is yes. The idea is as follows. A basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span> is given by <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/o.f5nkkZC1PEov1Bjz5CUC3EQ.MnIDCsxksltQ.svgz" alt="1" title="1"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. Hence, we defined <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="15" src="https://math.fontein.de/formulae/HrAqXYeHHrXVhKXGmmzt0zLNdcuPSWGAsTHxKw.svgz" alt="V_\C = V \oplus V" title="V_\C = V \oplus V"></span>, where the first <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> corresponds to 1 and the second to <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>: i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="18" src="https://math.fontein.de/formulae/648GWmPGDILBJgfh.Q6bLblFGDx4aa4.qkLD5Q.svgz" alt="(v, w) \in V_\C" title="(v, w) \in V_\C"></span> should mean <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="13" src="https://math.fontein.de/formulae/alrji..RwxDaY1nirDWyQp1ozrRAqM2wNjIdVQ.svgz" alt="v + i w" title="v + i w"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/eRFGzdgo63Xz4sTFD6lWtCyiS6HvojruYdD_Gg.svgz" alt="\F_8 / \F_2" title="\F_8 / \F_2"></span> has a basis with three elements, so one could define <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="15" src="https://math.fontein.de/formulae/MR3R089jOWS88I_N66rDqpB_u8KiAk7l2cvXLg.svgz" alt="V_L := V \oplus V \oplus V" title="V_L := V \oplus V \oplus V"></span>. And for <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/8n6RrwDiwZhvnfxvtHIicfgMKXfTdGwB_aTtNg.svgz" alt="V_L" title="V_L"></span> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="15" src="https://math.fontein.de/formulae/UxlQfHmMNpMZSmsw4V0Yyc1xcO8HzcFZaHEI7g.svgz" alt="K = \Q" title="K = \Q"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="12" src="https://math.fontein.de/formulae/tdWnrYR0uO8hdTrPgen62JDFpJhc30uAGQkrKw.svgz" alt="L = \C" title="L = \C"></span>, we need an infinite basis and an infinite direct sum.
</p>
<p>
It would be nice if we could avoid working with bases, both of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and of the field extension <span class="inline-formula"><img class="img-inline-formula img-formula" width="37" height="18" src="https://math.fontein.de/formulae/2X12W49lj02TDhOLBRCb3NHtYkwNtYH0Fovx6A.svgz" alt="L/K" title="L/K"></span>. This can indeed be done, using the <em>tensor product</em>. We begin with a very abstract defintion.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> be a ring and <span class="inline-formula"><img class="img-inline-formula img-formula" width="38" height="16" src="https://math.fontein.de/formulae/pYqBeibDl81KrrYt29WuBTEFkP30mZPojTJ62Q.svgz" alt="V, W" title="V, W"></span> <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-modules. A pair <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/dByug3n18gLuC1f3XMw9Fh7B72RkzHSmj2NLDg.svgz" alt="(T, \phi)" title="(T, \phi)"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-module and <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="16" src="https://math.fontein.de/formulae/7dDtM2Bic9l1ncXMfQ5Oh_MvcoQIPiaXUJhYEw.svgz" alt="\phi : V \times W \to T" title="\phi : V \times W \to T"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-bilinear, is said to be a <em>tensor product</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> if the following <em>universal property</em> holds:
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> is any <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-module and <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="16" src="https://math.fontein.de/formulae/V8n6m.gOeUSYaemMAybtGiMyYYLt45uX_mEObw.svgz" alt="\psi : V \times W \to A" title="\psi : V \times W \to A"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-bilinear, there exists exactly one homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/OJj8IkaUtYY5XnlZCNDMBmlZVY819VZdF8qhMA.svgz" alt="\varphi : T \to A" title="\varphi : T \to A"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="16" src="https://math.fontein.de/formulae/JKKDE4aRmBM_tzKQOqN2IQoLYWjeQ2QmIiXyAA.svgz" alt="\psi = \varphi \circ \phi" title="\psi = \varphi \circ \phi"></span>.
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="152" height="97" src="https://math.fontein.de/formulae/KUhtqfZwTCwz4jLvfvdecSQLHA4hl0P4HBnkkQ.svgz" alt="\xymatrix{ V \times W \ar[r]^\phi \ar[rd]_\psi & T \ar@{-->}[d]^{\exists! \varphi} \\ & A }" title="\xymatrix{ V \times W \ar[r]^\phi \ar[rd]_\psi & T \ar@{-->}[d]^{\exists! \varphi} \\ & A }">
</div>
</div>
</div>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Tensor products exist and are unique up to unique isomorphism. More precisely, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/dByug3n18gLuC1f3XMw9Fh7B72RkzHSmj2NLDg.svgz" alt="(T, \phi)" title="(T, \phi)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/8W_sNP53LzQdC.1KHqKf3akPtrCZcOcVdX5InQ.svgz" alt="(T', \phi')" title="(T', \phi')"></span> are tensor products of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>, there exists exactly one <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-isomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="17" src="https://math.fontein.de/formulae/GON2Fpi0gQYHRq1nhWxa8nj85unPh5mBEv2riw.svgz" alt="\varphi : T \to T'" title="\varphi : T \to T'"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="17" src="https://math.fontein.de/formulae/mbn77UQ4JInwZULH.rozF9Esa13JKcmsIgreog.svgz" alt="\varphi \circ \phi = \phi'" title="\varphi \circ \phi = \phi'"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
From now on, we write <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/uea7mB4EHBD581fe5RuWrlAj4I01OsrvkFrAYg.svgz" alt="V \otimes_R W" title="V \otimes_R W"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/NlJVsDKqqV5jBHHEceVtse6TyXCHH9OQyWen_Q.svgz" alt="T" title="T"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="13" src="https://math.fontein.de/formulae/Q4_cNI4WkAauRtjA4OAurt1wpf5WZdL1l4okFQ.svgz" alt="v \otimes_R w" title="v \otimes_R w"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/_ZPWATfanu4xnM8Or15z7v7ALN7.TRfHtiXagQ.svgz" alt="\phi(v, w)" title="\phi(v, w)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/XSbOyihEe.R6HcS05dfEgM5nTKitkXpxe_Kwxg.svgz" alt="v \in V" title="v \in V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="13" src="https://math.fontein.de/formulae/t6ueWtWakzOc_bRQuGKpq7NWCrXp9Boi3AQWsQ.svgz" alt="w \in W" title="w \in W"></span>. In case the base <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span> is clear, we will drop the subscript.
</p>
<p>
As we are interested in tensor products of vector spaces over a field, we can be more concrete.
</p>
<div class="theorem-environment theorem-theorem-environment qed">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> be <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector spaces. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="18" src="https://math.fontein.de/formulae/oj9Aog9a3HJ5YRMYgI6v52UDUjEka9FUI_UbfA.svgz" alt="(v_i)_{i\in I}" title="(v_i)_{i\in I}"></span> be a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="60" height="18" src="https://math.fontein.de/formulae/j0qoo5hFRsrQppnrjzNkXEljftaVzCP3V7fexA.svgz" alt="(w_j)_{j\in J}" title="(w_j)_{j\in J}"></span> be a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="20" src="https://math.fontein.de/formulae/Q.nstFyi7ZVtpcEp7bXsvyZKAya6QutxMNK5Zw.svgz" alt="(v_i \otimes w_j)_{(i, j) \in I \times J}" title="(v_i \otimes w_j)_{(i, j) \in I \times J}"></span> is a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="15" src="https://math.fontein.de/formulae/Xd0ND1kptzQyzYyBaIa2_ny9sQ1OE8Db.IrdVQ.svgz" alt="V \otimes_K W" title="V \otimes_K W"></span>. In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="288" height="18" src="https://math.fontein.de/formulae/ODsr3vZD58MfBT9PeY9dGouOg.XthvKerIzt9g.svgz" alt="\dim_K (V \otimes_K W) = \dim_K V \cdot \dim_K W" title="\dim_K (V \otimes_K W) = \dim_K V \cdot \dim_K W"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
A different interpretation is that <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/uea7mB4EHBD581fe5RuWrlAj4I01OsrvkFrAYg.svgz" alt="V \otimes_R W" title="V \otimes_R W"></span> is the set of linear combinations of elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span>, where the coefficients are elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>. Hence, we extend the range of the coefficients of elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> from <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>. Every element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/uea7mB4EHBD581fe5RuWrlAj4I01OsrvkFrAYg.svgz" alt="V \otimes_R W" title="V \otimes_R W"></span> can be written in the form <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="20" src="https://math.fontein.de/formulae/Vv87Z9XMZNN2IguYryPPDcoccBY3D6RQ7Nh.uw.svgz" alt="\sum_{i=1}^n v_i \otimes w_i" title="\sum_{i=1}^n v_i \otimes w_i"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="15" src="https://math.fontein.de/formulae/cF91htNEfUQ12frpmGaRzkPRpnK0CsfcFofYLA.svgz" alt="v_i \in V" title="v_i \in V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="15" src="https://math.fontein.de/formulae/OKNMRRRrjLqeEcAATCxJC9CczERC8QAjtYnhUA.svgz" alt="w_i \in W" title="w_i \in W"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="14" src="https://math.fontein.de/formulae/FpZ4n1WCX1xSEeoMKKG9Jn.6P1n9K4u5Fncztw.svgz" alt="1 \le i \le n" title="1 \le i \le n"></span>.
</p>
<p>
Now let <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span> be a field extension of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector space, whence we can consider the tensor product <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="15" src="https://math.fontein.de/formulae/9gepX23xynrCvkg83R1QB6aYesrWGF2qRfPg8w.svgz" alt="V_L := L \otimes_K V" title="V_L := L \otimes_K V"></span>. As expected, this turns out to be a <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>-vector space with scalar multiplication <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="15" src="https://math.fontein.de/formulae/mGCG1rmEG4l6Osi8lvGICL.Ln3bZ7cGOTnTiNg.svgz" alt="\C \times V_L \to V_L" title="\C \times V_L \to V_L"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="276" height="20" src="https://math.fontein.de/formulae/jWM9a.Q8HCjwTMCIpU_88rN8FW5GZvboG2gp2Q.svgz" alt="(\lambda, \sum_{i=1}^n \lambda_i \otimes v_i) \mapsto \sum_{i=1}^n (\lambda \lambda_i) \otimes v_i" title="(\lambda, \sum_{i=1}^n \lambda_i \otimes v_i) \mapsto \sum_{i=1}^n (\lambda \lambda_i) \otimes v_i"></span>. In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="15" src="https://math.fontein.de/formulae/k2CBuEYawNL350Oz460u8IizJdI7Pq9kEci_RQ.svgz" alt="\lambda \in K \subseteq L" title="\lambda \in K \subseteq L"></span>, this definition coincides with the natural <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector space structure of <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/8n6RrwDiwZhvnfxvtHIicfgMKXfTdGwB_aTtNg.svgz" alt="V_L" title="V_L"></span>.
</p>
<p>
Let us consider the special case <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/6LMDm8s8_La_eisR.EwfW4CG3W0YM2lSNqLc_Q.svgz" alt="K = \R" title="K = \R"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="12" src="https://math.fontein.de/formulae/tdWnrYR0uO8hdTrPgen62JDFpJhc30uAGQkrKw.svgz" alt="L = \C" title="L = \C"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="37" height="18" src="https://math.fontein.de/formulae/taEv7yQmWP1S9okiUcEf4jUpiGFGlwc_8oIuzQ.svgz" alt="(1, i)" title="(1, i)"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>; if <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="18" src="https://math.fontein.de/formulae/yTYhYv4PlvOQk1RuNu4e6bi4IxqTaL8F0JnpWw.svgz" alt="(v_j)_{j\in J}" title="(v_j)_{j\in J}"></span> is an <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="18" src="https://math.fontein.de/formulae/rgoAD_NHLEMvVJ0QBwAOBRqmeqDWv7VWtcM.iw.svgz" alt="(v_j, i v_j)_{j \in J}" title="(v_j, i v_j)_{j \in J}"></span> is an <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span>: every element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span> can be written in the form <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="13" src="https://math.fontein.de/formulae/alrji..RwxDaY1nirDWyQp1ozrRAqM2wNjIdVQ.svgz" alt="v + i w" title="v + i w"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="16" src="https://math.fontein.de/formulae/f_wEqf3.bvZ9b.Lt2NmngW.okfRdCISJGI2l3Q.svgz" alt="v, w \in V" title="v, w \in V"></span>. Moreover, <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="18" src="https://math.fontein.de/formulae/yTYhYv4PlvOQk1RuNu4e6bi4IxqTaL8F0JnpWw.svgz" alt="(v_j)_{j\in J}" title="(v_j)_{j\in J}"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span>. Compare this with the ad-hoc definition of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/M05fkJ.QvxnMuj3L3.xOcVu6jim0moF7SOKKKw.svgz" alt="V_\C" title="V_\C"></span> at the beginning of this post.
</p>
<p>
Now, let us consider what to do with <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-linear maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="16" src="https://math.fontein.de/formulae/kpf9BVs5I.KiR48TJFX9fT7XIz2sGEsJcG9o_w.svgz" alt="\phi : V \to W" title="\phi : V \to W"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> are <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span>-vector spaces. We begin with a general result on tensor products.
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="16" src="https://math.fontein.de/formulae/Oa.cTgQd3R8OFlK3Y9Oy77mLhJL0tB68PSlucA.svgz" alt="V_i, W_i" title="V_i, W_i"></span> be <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-modules, <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="15" src="https://math.fontein.de/formulae/TdwOk561h7dDPcmtosIs36ZnSI2.45Gl5fJdRQ.svgz" alt="i = 1, 2" title="i = 1, 2"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="16" src="https://math.fontein.de/formulae/5poArkaDOa3r1L4mzvytzVkzqshlEq.DmkCkwA.svgz" alt="\phi_i : V_i \to W_i" title="\phi_i : V_i \to W_i"></span> be <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-module homomorphisms. Then there exists exactly one <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="16" src="https://math.fontein.de/formulae/9VQufJ4ANNMfJzVuGyBsqrH4sKKEVLZNuKQn4Q.svgz" alt="\phi : V_1 \otimes V_2 \to W_1 \otimes W_2" title="\phi : V_1 \otimes V_2 \to W_1 \otimes W_2"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="222" height="18" src="https://math.fontein.de/formulae/a.OjEGDycDlm9nu31NaM_Tgq29wEh6hxe_SePQ.svgz" alt="\phi(v_1 \otimes v_2) = \phi_1(v_1) \otimes \phi_2(v_2)" title="\phi(v_1 \otimes v_2) = \phi_1(v_1) \otimes \phi_2(v_2)"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="113" height="15" src="https://math.fontein.de/formulae/3RuYR8p8_cXg3p1tphfnTYa8TotUIz4fucEJYg.svgz" alt="A := W_1 \otimes W_2" title="A := W_1 \otimes W_2"></span> and define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="358" height="18" src="https://math.fontein.de/formulae/ewe1ilx9QtDVhTyiWoRZuQyRgAZgSS2U0oQ_yw.svgz" alt="\psi : V_1 \times V_2 \to A, \quad (v_1, v_2) \mapsto \phi_1(v_1) \otimes \phi_2(v_2)." title="\psi : V_1 \times V_2 \to A, \quad (v_1, v_2) \mapsto \phi_1(v_1) \otimes \phi_2(v_2).">
</div>
<p>
One quickly checks that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="16" src="https://math.fontein.de/formulae/5YQhZpth_qEZfoZavf10MlpXOOEYGw_G8IF6hw.svgz" alt="\psi" title="\psi"></span> is bilinear. Hence, by the definition of the tensor product <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="15" src="https://math.fontein.de/formulae/K9kz0oh1u81MLC6PGEbkiYERNO8aKyadrsaaDA.svgz" alt="V_1 \otimes V_2" title="V_1 \otimes V_2"></span>, there exists exactly one <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="124" height="16" src="https://math.fontein.de/formulae/cXvJ__2B8uQ4l9iIIdCoNtmV2lnFRCbVCy2UCg.svgz" alt="\phi : V_1 \otimes V_2 \to A" title="\phi : V_1 \otimes V_2 \to A"></span> with
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="317" height="18" src="https://math.fontein.de/formulae/CyQCIg21xKpVNVuDJqGkv8Fho8lZnE031ZVsHQ.svgz" alt="\phi(v_1 \otimes v_2) = \psi(v_1, v_2) = \phi_1(v_1) \otimes \phi_2(v_2)." title="\phi(v_1 \otimes v_2) = \psi(v_1, v_2) = \phi_1(v_1) \otimes \phi_2(v_2).">
</div>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now let us consider <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector spaces <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span>, a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear map <span class="inline-formula"><img class="img-inline-formula img-formula" width="87" height="16" src="https://math.fontein.de/formulae/svMeGa.P7s7KK1TA2ZmdCCd_WJ9.4hityl.yGg.svgz" alt="\varphi : V \to W" title="\varphi : V \to W"></span> and the identity map <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="15" src="https://math.fontein.de/formulae/XliFR_3OAfO4KPXs2BVdPhj8L.4fjc3zuo6YCQ.svgz" alt="\id_L : L \to L" title="\id_L : L \to L"></span>. By the theorem, there exists exactly one <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="288" height="16" src="https://math.fontein.de/formulae/XjU9YExWTTAoK5IaoWOPIEdNhbyNfHe54_TRvw.svgz" alt="\varphi_L : V_L = L \otimes_K V \to L \otimes_K W = W_L" title="\varphi_L : V_L = L \otimes_K V \to L \otimes_K W = W_L">
</div>
<p>
with <span class="inline-formula"><img class="img-inline-formula img-formula" width="206" height="18" src="https://math.fontein.de/formulae/jejQ..sfwoneIjP6zCpD0uAvEW9PscNR7bvKgQ.svgz" alt="\varphi_L(\lambda \otimes v) = \id_L(\lambda) \otimes \varphi(v)" title="\varphi_L(\lambda \otimes v) = \id_L(\lambda) \otimes \varphi(v)"></span>. But since <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="14" src="https://math.fontein.de/formulae/2K_A.C172tR5ci3TMGJFdpSKoA.dKJunMlqZnQ.svgz" alt="\lambda \otimes v" title="\lambda \otimes v"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="12" src="https://math.fontein.de/formulae/8nUBhurVBj68zhlweME21wMzNdX0n_qrLHJrSA.svgz" alt="\lambda v" title="\lambda v"></span>, using the <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>-vector space structure of <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/8n6RrwDiwZhvnfxvtHIicfgMKXfTdGwB_aTtNg.svgz" alt="V_L" title="V_L"></span>, we obtain <span class="inline-formula"><img class="img-inline-formula img-formula" width="135" height="18" src="https://math.fontein.de/formulae/HeMUKEmC_T4LMtkCuYliPrZeZVYZvWZZy5R2Pw.svgz" alt="\varphi_L(\lambda v) = \lambda \varphi_L(v)" title="\varphi_L(\lambda v) = \lambda \varphi_L(v)"></span>, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="11" src="https://math.fontein.de/formulae/qsGWXQ7H2ZRceOw2Gy_h_bKpnWN8NpMpZe_TmQ.svgz" alt="\varphi_L" title="\varphi_L"></span> is <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>-linear.
</p>
<p>
Finally, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="89" height="18" src="https://math.fontein.de/formulae/zA2p9esgeKt7ZIWH6nTCplI6FMGDyBQ0jIT6CA.svgz" alt="B = (v_i)_{i\in I}" title="B = (v_i)_{i\in I}"></span> be a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="103" height="18" src="https://math.fontein.de/formulae/DHV2uvINosG5kmOzCfQeqGJ0KXt.pqmITn34sw.svgz" alt="B' = (w_j)_{j\in J}" title="B' = (w_j)_{j\in J}"></span> be a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/Ih7V6Mp4twAjwq7Tr86Dt9U5c5VQJd0XWRPlqw.svgz" alt="B" title="B"></span> is as well a <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/8n6RrwDiwZhvnfxvtHIicfgMKXfTdGwB_aTtNg.svgz" alt="V_L" title="V_L"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="13" src="https://math.fontein.de/formulae/9huHplcfo.6T56TKdyIFi.JPZxW7dKIeWciduw.svgz" alt="B'" title="B'"></span> is as well a <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/.Du.FEff5VhxA7vvy65XSXJ9V005fsu_zcw6TQ.svgz" alt="L" title="L"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span>, whence we can consider the matrices <span class="inline-formula"><img class="img-inline-formula img-formula" width="141" height="20" src="https://math.fontein.de/formulae/RS94pWhCVsz1k4W1yfjWwmpoGQ_YrTk47OUOow.svgz" alt="M_{B,B'}(\varphi) \in K^{J \times I}" title="M_{B,B'}(\varphi) \in K^{J \times I}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="148" height="20" src="https://math.fontein.de/formulae/6g3GGMt6JDcS2EvFxaw.jdwEXujyNc5_y7FpbQ.svgz" alt="M_{B,B'}(\varphi_L) \in L^{J \times I}" title="M_{B,B'}(\varphi_L) \in L^{J \times I}"></span>. Write <span class="inline-formula"><img class="img-inline-formula img-formula" width="154" height="21" src="https://math.fontein.de/formulae/.OaBcPny4fovjEk1sN8pFHNuUoeHSxnBjPqAxQ.svgz" alt="\varphi(v_i) = \sum_{j\in J} \lambda_{ij} w_j" title="\varphi(v_i) = \sum_{j\in J} \lambda_{ij} w_j"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="25" src="https://math.fontein.de/formulae/sw.ZKbIkxsb.ncmAwiN2Rm_gsp34qamUJZIeJw.svgz" alt="M_{B,B'}(\varphi) = (\lambda_{ij})_{i \in I, \atop j \in J}" title="M_{B,B'}(\varphi) = (\lambda_{ij})_{i \in I, \atop j \in J}"></span>. Now
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="415" height="69" src="https://math.fontein.de/formulae/uvEdRh_BLpf1WXfLI.Hl11Re.A6rz7LjXCUYKQ.svgz" alt="\varphi_L(v_i) ={} & \varphi_L(1 \otimes_K v_i) = \id_L(1) \otimes_K \varphi(v_i) \\
{}={} & \id_L(1) \otimes_K \sum_{j\in J} \lambda_{ij} w_j = \sum_{j\in J} \lambda_{ij} (\id_L(1) \otimes w_j)." title="\varphi_L(v_i) ={} & \varphi_L(1 \otimes_K v_i) = \id_L(1) \otimes_K \varphi(v_i) \\
{}={} & \id_L(1) \otimes_K \sum_{j\in J} \lambda_{ij} w_j = \sum_{j\in J} \lambda_{ij} (\id_L(1) \otimes w_j).">
</div>
<p>
Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="273" height="25" src="https://math.fontein.de/formulae/lFCAPqg_.2wRT6XuLYMlBgK2x2fZsOIdlZkuqQ.svgz" alt="M_{B,B'}(\varphi_L) = (\lambda_{ij})_{i \in I, \atop j \in J} = M_{B,B'}(\varphi)" title="M_{B,B'}(\varphi_L) = (\lambda_{ij})_{i \in I, \atop j \in J} = M_{B,B'}(\varphi)"></span> as well.
</p>
<p>
Hence, the tensor product allows us to describe <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/8n6RrwDiwZhvnfxvtHIicfgMKXfTdGwB_aTtNg.svgz" alt="V_L" title="V_L"></span>, as a generalization of the complexification of real vector spaces, in a very clean and abstract manner.
</p>
<p>
Finally, recall that every field <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> has an algebraical closure <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/9cRej4VZYxmVLYJhLZclFdOBm1_XNssdEY3tOg.svgz" alt="\overline{K}" title="\overline{K}"></span>, which is unique up to <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-isomorphism. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-vector spaces <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-linear maps <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="16" src="https://math.fontein.de/formulae/kpf9BVs5I.KiR48TJFX9fT7XIz2sGEsJcG9o_w.svgz" alt="\phi : V \to W" title="\phi : V \to W"></span> we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/9cRej4VZYxmVLYJhLZclFdOBm1_XNssdEY3tOg.svgz" alt="\overline{K}" title="\overline{K}"></span>-vector spaces <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="17" src="https://math.fontein.de/formulae/dPmPp4SnUNPZ9IBb174K.q1UFIH1WZyHqY8AXg.svgz" alt="V_{\overline{K}}" title="V_{\overline{K}}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="30" height="17" src="https://math.fontein.de/formulae/WimfZNvYV0jazTSiFYJA_dzcUsxfiRPDTwKbKQ.svgz" alt="W_{\overline{K}}" title="W_{\overline{K}}"></span> and a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/9cRej4VZYxmVLYJhLZclFdOBm1_XNssdEY3tOg.svgz" alt="\overline{K}" title="\overline{K}"></span>-linear map <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="18" src="https://math.fontein.de/formulae/60hJtdqyXolTTslygF3SaTMXkQdMkIqBTiO3jA.svgz" alt="\phi_{\overline{K}} : V_{\overline{K}} \to W_{\overline{K}}" title="\phi_{\overline{K}} : V_{\overline{K}} \to W_{\overline{K}}"></span>. We have seen that every <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span> resp. <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/Et.q96aMad6BWAg3tHEglwYQth04VLyi7uR4lA.svgz" alt="W" title="W"></span> is also an <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/9cRej4VZYxmVLYJhLZclFdOBm1_XNssdEY3tOg.svgz" alt="\overline{K}" title="\overline{K}"></span>-basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="17" src="https://math.fontein.de/formulae/dPmPp4SnUNPZ9IBb174K.q1UFIH1WZyHqY8AXg.svgz" alt="V_{\overline{K}}" title="V_{\overline{K}}"></span> resp. <span class="inline-formula"><img class="img-inline-formula img-formula" width="30" height="17" src="https://math.fontein.de/formulae/WimfZNvYV0jazTSiFYJA_dzcUsxfiRPDTwKbKQ.svgz" alt="W_{\overline{K}}" title="W_{\overline{K}}"></span>, and that the matrix representation of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span> with respect to the bases equals the one of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="18" src="https://math.fontein.de/formulae/7bRbCoC7KQGoNuzoZSA5O.t8.i8S47nF9UpRyA.svgz" alt="\phi_{\overline{K}}" title="\phi_{\overline{K}}"></span>. Hence, we can not just talk of arbitrary elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="16" src="https://math.fontein.de/formulae/9cRej4VZYxmVLYJhLZclFdOBm1_XNssdEY3tOg.svgz" alt="\overline{K}" title="\overline{K}"></span> being eigenvalues of matrices <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="18" src="https://math.fontein.de/formulae/XOWnwo7s7MpJhqowl6x.UZ0KFKoFUHf78jvdlQ.svgz" alt="M_{B,B'}(\phi)" title="M_{B,B'}(\phi)"></span> over <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, but also of endomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/nnhzdqchKKCzaVcrUWETL.T75aNctBowFpJaFA.svgz" alt="\phi" title="\phi"></span> defined over <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, by referring to <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="19" src="https://math.fontein.de/formulae/lm6Wcw.GcYVZFI9dHDemO34j8rDsyyXVhT3qCw.svgz" alt="M_{B,B'}(\phi_{\overline{K}})" title="M_{B,B'}(\phi_{\overline{K}})"></span> resp. <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="18" src="https://math.fontein.de/formulae/7bRbCoC7KQGoNuzoZSA5O.t8.i8S47nF9UpRyA.svgz" alt="\phi_{\overline{K}}" title="\phi_{\overline{K}}"></span> instead.
</p>
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