Isomorphisms between dual vector spaces

Question

Let X and Y be k-vector spaces, if they are finite dimensional and $D=Hom_k(-,k)$ we have a natural (in X and Y) isomorphism:

$$ D Hom_k(X,Y) \cong Hom_k(DY,DX) $$

Can anyone help me prove this? I already tried to give the explicit isomorphism, but failed.

Another question, can we replace the statement for X and Y finite dimensional A-modules for a finite dimensional k-algebra A? Namely:

$$ D Hom_A(X,Y) \cong Hom_{A^{op}}(DY,DX) $$

Is it still natural in X and Y?

Answer 1

I think that this cannot be true. Via passing to the dual map, $L(X,Y)$ is naturally isomorphic to $L(Y^*,X^*)$ (for finite dimensional $X$ and $Y$). If what you claim were true, then $L(X,Y)$ would be naturally isomorphic to $L(X,Y)^*$ which for $X=k$ would give you a natural isomorphism between $Y$ and $Y^*$.

In your orginal setup things alos do not fit together since one of the two functors you are looking at is covariant in $X$ and contravariant in $Y$ while the other is contravariant in $X$ and covaraint in $Y$.