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?



$k$ structures on $K$ vector spaces
The statement appears to be wrong. What you need is for $V$ to be finitely generated. It is a general theorem of commutative algebra that, if $R\rightarrow S$ is a flat map of commutative rings, and $M$ and $N$ are $R$-modules with $M$ finitely presented over $R$, then the natural map
$$Hom\_R(M,N)\rightarrow Hom\_S(S\otimes\_RM,S\otimes\_RN)$$
induces an isomorphism
$$Hom\_R(M,N)\otimes\_RS\rightarrow Hom\_S(S\otimes\_RM,S\otimes\_RN).$$
See Eisenbud, Proposition 2.10.


If we take $R=k$ and $S=K$, you get the result about $k$-structures as long as $V$ is finitely generated.


