Background:
Let $V$ be a vector space over a field $\Bbbk$. Let $V^*=Hom_\Bbbk(V,\Bbbk)$ be the vector space of linear functionals on $V$. Consider the map $V^*\times V\to End_\Bbbk(V)$ given by $(f,v_0)\to\phi$ where $\phi(v)=f(v)\cdot v_0$. This map is obviously bilinear and thus induces a linear map $V^*\otimes_\Bbbk V\to End_\Bbbk V$.
If $V$ is finite-dimensional, then a choice of basis $\{e_1,\dots,e_n\}$ and a dual basis $\{e_1^*,\dots,e_n^*\}$ establishes that the map is injective since evaluations at $e_i$ for the image of each basis element $e_i^*\otimes e_j$ of $V^*\otimes V$ shows that the corresponding endomorphisms are linearly independent. Identifying $End_\Bbbk(V)$ with the $n^2$-dimensional space of $n\times n$ matrices and the fact that $\dim V^*\otimes V$ is $n^2$ shows that the map is in fact an isomorphism.
Now onto my question. If $V$ is an infinite-dimensional vector space over $\Bbbk$, then:
- is the above (obvious?) map $V^*\otimes_\Bbbk V\to End_\Bbbk(V)$ still injective? If not, then under what conditions is it?
- given a $\phi\in End(V)$, is there a criterion to determine whether $\phi$ comes from an element of $V^*\otimes V$?