Let $V$ be a vector space (not necessarily finite-dimensional) over a field $\mathbb F$. Let $B(V^*)$ denote the vector space of all bilinear forms $V^*\times V^* \rightarrow \mathbb F$. It is "well-known" that the linear map $\phi: V\otimes V \rightarrow B(V^*)$ such that $$\phi(v \otimes w)(f,g) = f(v)g(w)$$ for all $v,w\in V$ and $f,g\in V^*$ establishes an isomorphism $V\otimes V \cong B(V^*)$ when $V$ is finite-dimensional. I want to prove that this hyphotesis cannot be dropped, that is, I'm looking forward a vector space $V$ of infinite dimension such that $V\otimes V \not \cong B(V^*).$
I am not familiar at all with infinite-dimensional vector spaces, besides the classical ones $\mathbb F[x]$ and $\mathscr C([0,1],\mathbb R)$, for instance. Which vector space should I look for? Is this problem more related to the cardinality of basis, in the sense that $\dim B(V^*)$ is also uncountable if $\dim V^*$ is uncountable? Any help is very much appreciated.