Example of $V^* \otimes V^*$ not isomorphic to $(V \otimes V)^*$

The question that makes sense is:

if $V$ is infinite dimensional, is the canonical map $T:V^*\otimes V^*\to (V\otimes V)^*$ surjective?

and the answer is no. To see this we have to find something about the elements of the image of $T$ which is special.

Suppose that $\phi\in(V\otimes V)^*$ and consider the set $U(\phi)$ of all vectors $u\in V$ such that $\phi(u\otimes v)=0$ for all $v\in V$. This is a subspace of $V$.

If $a\in V^*\otimes V^*$, so that there are $n\geq0$ and $\phi_1,\dots,\phi_n$, $\psi_1,\dots,\psi_n\in V^*$ such that $a=\sum_{i=1}^n\phi_i\otimes\psi_i$, then the space $K(a)=\bigcap_{i=1}^n\ker\phi_i$ has finite codimension in $V$ and is contained in $U(T(a))$. This tells us that

if $\phi\in(V\otimes V)^*$ is in the image of the map $T$, then $U(\phi)$ is a subspace of $V$ of finite codimension.

Now, to show that $T$ is not surjective it is enough that we exhibit a $\phi\in (V\otimes V)^*$ such that $U(\phi)$ is not of finite codimension!

Let $\{e_i\}_{i\in I}$ be a basis for $V$. There is a unique $\phi\in(V\otimes V)^*$ such that $$\phi(e_i\otimes e_j)=\begin{cases}1, & \text{if $i=j$;} \\0, & \text{if not.}\end{cases}$$ Let $u\in V$. As we have a basis, there are scalars $a_i$, one for each $i\in I$ and almost all of which are zero, such that $u=\sum_{i\in I}a_ie_i$. If $j\in I$ is such that $a_i\neq0$, then $\phi(u\otimes e_i)=a_i\neq0$, and we see that $u\not\in U(\phi)$ unless $u=0$. In other words, we have $U(\phi)=0$ and certainly the zero subspace of $V$ does not have finite codimension in $V$. The element $\phi$ is therefore not in the image of the map $T$.

In fact, we can reverse this. Suppose that $\lambda\in(V\otimes V)^*$ is such that $U(\lambda)$ has finite codimension in $V$. Let $n$ be that codimension, let $\{u_1,\dots,u_n\}$ be a basis of $V/U(\lambda)$ and let $\{\bar\phi_1,\dots,\bar\phi_n\}$ be the corresponding dual basis for the dual space of $V/U(\lambda)$. Let $p:V\to V/U(\lambda)$ be the canonical map and for each $i\in\{1,\dots,n\}$ let $\phi_i=\bar\phi_i\circ p\in V^*$. As $\lambda$ vanishes on the subspace $U(\phi)\otimes V$ of $V\otimes V$, it induces a map $\Lambda:(V/U(\lambda))\otimes V\to k$ (here $k$ is the field) For each $i\in\{1,\dots,n\}$ we consider the map $\psi_i\in V^*$ such that $\psi_i(v)=\Lambda(u_i\otimes v)$. After all this setup, one can easily check now that $\lambda=T(\sum_{i=1}^n\phi_i\otimes\psi_i)$.

We have this proved:

The image of the canonical map $V^*\otimes V^*\to(V\otimes V)^*$ is precisely the subset of all $\lambda\in(V\otimes V)^*$ such that $U(\lambda)$ has finite codimension in $V$.