Hom set as a vector space isomorphic to direct sum

Question

Let $V$ be a vector space of countable dimension over the field $K$. How to show as $K$-vector spaces, $\text{Hom}(V,V)$ is isomorphic to the direct sum of $V$ and $V$?

Answer 1

You can't because they are not. To easily see this, consider the special case that $K$ is finite (or countably infinite) and $\{v_i\}_{i\in\mathbb N}$ is a basis of $V$. Then $V$ is countable (as a set) and so is $V\oplus V$. But $\operatorname{Hom}(V,V)$ is not countable because for any subset $A$ of $\mathbb N$ we can definie a linear map $f\colon V\to V$ by $$f(v_i)=\begin{cases}v_i&\text{if }i\in A\\0&\text{otherwise}\end{cases}$$ and these are pairwise distinct.

Answer 2

Do you maybe want to show that $$Hom(V,V) \cong V^* \otimes V?$$ This result is quite nice. However, I have not checked if it holds for infinite diminsional vectorspaces.