The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.

Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is said to be defined over $k$ if $f(V_k)\subset W_k$ and these are elements of $Hom_K(V,W)_k\subset Hom_K(V,W)$. This is a $k$ structure if $W$ is finite dimensional.

The author seems to be making no assumption on the dimension of $V$ (which is the source of my problem). If we allow $V$ to be infinite dimensional then it seems to me that it is incorrect that $Hom_K(V,W)_k$ is a $k$ structure on $Hom_K(V,W)$ as is given by the following example. Let $V_k=\oplus_{i\geq0} ke_i$ and take $K$ to be an extension which is not a finite $k$ vector space, and define $f\in Hom_K(V,K)$ by choosing $f(e_i)$ which are linearly independent over $k$. We will never be able to write such an element as $\sum f_i\otimes \alpha_i$ with $f_i\in Hom_K(V,K)_k$.

So my question is : Do we need to assume both $V,W$ to be finite dimensional $K$ vector spaces for $Hom_K(V,W)$ to have a $k$ structure?

Also, what are the other references for $k$ structures and rationality properties?

Pseudo-reductive Groupsby Conrad-Gabber-Prasad may be the best modern reference for rationality properties. – Jim Humphreys Apr 21 '11 at 14:40