Let and be vector spaces over a field and , their duals. In case is finite dimensional, one obtains a non-canonical isomorphism , a canonical isomorphism and a canonical isomorphism .

In case , and are not isomorphic: a basis of has a cardinality strictly larger than the one of . Moreover, the canonical map is still a monomorphism, but no longer surjective. In the case of , one has as well a canonical monomorphism , but it is no longer surjective as well. We want to study the images of the canonical maps and .

We begin with an auxiliary lemma.

Choose a -basis of such that there exists some with . Define by . Then and .

Clearly, for , is -linear. Moreover, one quickly sees that is -linear itself. To see that is injective, let with . Now, by the lemma, there exists a with ; this shows that , whence . Therefore, an is injective.

Now, if , , whence . This shows that the image of is contained in the given set. Now assume that satisfies ; say, . Then for all , whence . By the Homomorphism Theorem, there exists a homomorphism such that

commutes. Now , whence . As (as ), is an isomorphism and we must have for some . But then, lies in the image of .

Finally, if , we saw that we have for any non-zero , with depending on . Since is injective, this shows that we must have .

This allows us to show that is surjective if, and only if, .

We have that is surjective if, and only if, .

First, if , we see that . Since is injective, it follows that is in fact an isomorphism.

Now assume that . It suffices to construct a hyperplane in with ; this defines an element of which is not contained in the image of by the above proposition. For that, chose a basis of (using Zorn's lemma). This defines a family of elements of by , . Let be the subspace of generated by the 's. If we would have , we could emply Zorn's lemma a second time to obtain a hyperplane with ; this would prove our claim.

Hence, we have to show that . Note that for , ; in particular, for every , only finitely many of the 's are non-zero. Hence, it makes sense to define , . We claim that in case : for that, note that is a linear independent set in , since for every linear combination , we get for every .

Note that in fact, the proof shows that is isomorphic to a -fold direct product of , while is isomorphic to a -fold direct sum of . In case , these are of the same dimension, but in case , they are not.

We continue with the canonical map .

The map

is a monomorphism and its image is

the -vector space of finite dimensional -homomorphisms .

One quickly sees that defines an element of , whence is well-defined and its image is contained in . Moreover, one quickly sees that is a homomorphism.

Let with , i.e. with for all . Without loss of generality, we can assume that our representation of satisfies that the 's are linearly independent. In that case, implies for all . But since this is true for all , it follows that for all . But then, . Therefore, , whence is injective.

Now let , and let be a basis of . Let be the projections with an for . Set . Then for all since for all ; therefore, . This shows that , whence we have equality.

Now is a -algebra, whence for , it makes sense to define . We are interested on how can be described in terms of and . This is resolved by the following result:

Let be -vector spaces. The map

is -linear and the following diagram commutes:

Let , and be the canonical maps. Since these are isomorphisms, we have to show that for , and , we have . For that, let . Then

what we had to show.

In particular, is a -algebra isomorphic to ; it posseses a if, and only if, .

Now consider transposition

Clearly, transposition is injective:

The map is -linear and injective.

It is clear that is -linear. To see that it is injective, let with . Let ; then for all , whence by the first lemma. But that means .

We show that transposition restricts to the subspaces of the homomorphism spaces of homomorphisms with finite-dimensional image.

Let . The map

is an isomorphism. In particular, .

Let . The map , is well-defined and a homomorphism as and . Now let with , i.e. with . But since is defined on , this means that . Hence, is injective.

Now let , i.e. there exists some with . Set ; then and . Therefore, is injective.

Finally, in case , we have and , whence . On the contrary, if , we have , whence .

Now we have seen that and in a canonical way, and we have the canonical monomorphism . We show that these maps behave nicely with transposition:

The map

is the unique homomorphism which makes the diagram

commuting.

Let and . Then for , and

for all and . Hence, , what we had to show.

Now consider double transposition, i.e.

and its finite-dimensional image restriction

The above shows that using the canonical isomorphisms and , we can describe double transpotition by the following commuting diagram:

If , we obtain a map

The map is -linear and injective.

First, if is fixed, for all , ; hence, . Now, if and , , we have

whence is -linear.

To see that is injective, let be such that . Let and ; since for all , we see that , but since this is the case for all we get .

Note that we have the following diagram:

Moreover, using the canonical embeddings and , we can define a map by , and a map by . It turns out that these map make the diagram commute:

The maps , and , are -linear and make the diagram

commute. In particular, is injective.

That and are -linear is clear. For the lower triangle, let ; we have to show that . For that, let and ; then

For the right triangle, let ; we have to show that . For that, let and ; then

Now note that is injective. We can use this to determine the image of . For example, for ,

the last equivalence follows from the first proposition. Unfortunately, this criterion does not really helps in practice.

In case anyone knows a better description of the image of or , I'd be happy to know.

## Comments.

How do you publish your commutative diagrams, I assume using XY-pic syntax, (and other complex LaTeX, e.g. the cases environment) in Wordpress?

None of the plugins / blogs I've seen handle such complex use-cases.

Dear Shane,

I took the standard Wordpress LaTeX plugin and modified it for my purposes. I mainly added support for the align environment and for pstricks though, using XYpic is already possible with the default version in case you use your own server for LaTeX formula generation and not the Wordpress server (i.e. you need latex installed and accessible on your server). In case you can use your own server you can specify additions to the preamble (like

\usepackage{xypic}and\usepackage{amsmath}) in the wp-latex options. (I'm not sure if amsmath is included by default, i.e. you can already use the cases environment without further changes; but it might just be that you can't.) With these, you can use any XYpic commands inside the wp-latex math environments.