Let $V$ and $W$ be (not necessarily finite-dimensional) vector spaces. Show that there is a natural isomorphism (meaning an isomorphism that can be described without reference to a basis) between the vector spaces $L(V, L(V, W))$ and $BL(V\times V, W)$.
(Note: here $BL(V \times V, W)$ means the space of bilinear maps $V\times V \to W$, which means that is $B \in BL(V\times V, W)$ and $v \in V$ then the maps $B(v, \cdot)_V\to W$ and $B(\cdot, v):V \to W$, defined by $B(v, \cdot)(w) := B(v, w)$ and $B(\cdot, v)(w) := B(w, v)$ respectively, are both linear.)
I do not even know where to begin.
What is an element $B\in L(V, L(V,W))$? It's a linear map that for each vector $v_1 \in V$ gives you a linear map $B_{v_1}$ from $V$ to $L$. What happens then if you use that linear map on a second vector $v_2 \in V$? It gives you $B_{v_1}(v_2) \in W$. So, for any pair of vectors $v_1, v_2 \in V$ (or, we might say, for any vector $(v_1, v_2) \in V \times V$), we get a vector $B_{v_1}(v_2) \in W$. That means that $B$ seems to operate just as an element of $BL(V\times V, W)$ would.