# Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \in V \times W$. In the finite dimensional case it is easy to see this is an isomorphism by comparing dimensions on both sides. I’m aware that this is not an isomorphism in the infinite-dimensional case (see Relation with the dual space in the wikipedia article on tensor products) but I must admit that I’m not sure why. Cardinal arithmetic does not seem help here and, even if it did, I would be much happier to see an explicit example of a functional in $(V \otimes W)^*$ outside the range of this map. I’m having trouble cooking one up myself.

Added: Let me elaborate on my comment about cardinal arithmetic. Suppose that $X$ and $Y$ are infinite dimensional vector spaces over a field $k$. I’m confident that $\dim (X \otimes Y) = \dim X \cdot \dim Y$ holds. It is clear that $|X^*| = |k|^{\dim X}$ since we may identify a functional on $X$ with a function from a basis for $X$ to $k$. Also I think I’ve convinced myself that if $\dim X \geq |k|$ then in fact $|X| = \dim X$ ie. the cardinality and dimension of a vector space agree when the dimension is larger than the cardinality of the ground field. Consequently, assuming say that $|k| \leq \dim X \leq \dim Y$ it seems we have

which implies that in fact $(X \otimes Y)^*$ and $X^* \otimes Y^*$ are isomorphic but, rather frustratingly, the obvious map does not do the job. Did I make a mistake here? If not, is it generally true (ie without making assumptions about $|k|$) that there exists some isomorphism $(X \otimes Y)^* \to X^* \otimes Y^*$?

The map is not an isomorphism because an element in $X^*\otimes Y^*$ is a finite
sum of functionals of the form $x^{*}\otimes y^{*}$, where $x^* \in X^*$ and $y^* \in Y^*$. However, when $X$ and $Y$ are infinite dimensional, not every functional on $X\otimes Y$ will be of this form.

One case to consider, which will make this clear, is the case when $Y = X^*$.
There is one obvious element of $(X\otimes Y)^*$, namely the evaluation map
which takes a tensor $x\otimes y$ to the value of the functional $y$ on the vector $x$. Now one can check that this element is not in the image
of the map from $X^*\otimes Y^*$.

I am going to take a little time to rewrite the preceding example in a different language, because I think that it helps illustrate what is going on.

First note that if $Y = X^*$, then
$X\otimes Y$ embeds into $End(X)$
(the space of linear operators from $X$ to itself) as the space of finite rank linear operators (i.e. those whose image is finite dimensional). Denote
this image by $FREnd(X)\subset End(X)$. Note that any element of $FREnd(X)$
has a well-defined trace (because even though the domain is infinite dimensional, the range is finite dimensional); in the tensor product
description, this is just the natural map from $X\otimes Y$ to the ground field
given by evaluation of the functionals in $Y$ on the vectors in $X$.
(This is precisely the functional on $X\otimes Y$ that we considered above,
reexpressed in the language of operators.)

Replacing $X$ by $X^*$ in the preceding paragraph, we see that
$X^*\otimes Y^* = FREnd(X^*)$. Note that there is a natural map
$FREnd(X) \to FREnd(X^*)$ given by mapping an endomorphism $\phi$ to its
transpose $\phi^t$. (In terms of tensor products, this is the natural
map the isomorphism being the canonical one which switches the
two factors.)

The map $X^*\otimes Y^*\to (X\otimes Y)^*$
can then be reintrepreted as the pairing between
$FREnd(X^*)$ and $FREnd(X)^*$
defined as follows:
for $\phi\in FREnd(X^*)$ and $\psi \in FREnd(X),$

And now we see why this map is not surjective: for example, if
$\phi$ is any endomorphism of $X^*$, i.e. any element of $End(X^*)$,
then the composite $\phi\circ \psi^t$ has finite rank (since $\psi^t$ does),
and so $trace(\phi\circ\psi^t)$ is defined.

Thus we in fact have an embedding of all of $End(X^*)$ into $FREnd(X)^*$. With a little more work you can check that this latter embedding is an isomorphism. The conclusion in this case is that the embedding $X^*\otimes Y^* \to (X\otimes Y)^*$ can be reinterpreted as the embedding

which is not surjective when $X^*$ (or equivalently, $X$) is infinite dimensional, since it does not contain the identity map (for example).

Note that under the identification of $End(X^*)$ with $FREnd(X)^*$,
the identity map is identified precisely with the trace on $FREnd(X)$,
and so we get a reinterpretation of our original example, and see more clearly
what is going on: the point is that the identity endomorphism of an infinite dimensional vector space does not have finite rank.