Fact: vector spaces $BL(X,Y;Z)$ (set of bilinear maps from $X \times Y$ to $Z$) and $L(X, L(Y,Z))$ (set of linear maps from $X$ to $L(Y,Z)$) are isomorphic.
I know how to show two vector spaces are isomorphic: if there exists an invertible linear map between them, then two vector spaces are isomorphic .
My question is that how do I proceed to establish an 1-1 connection between the elements of sets of bilinear maps and linear maps (as mentioned in above fact). How do I feel naturally about the connection of elements of two sets, is there any systematic way to develop my thinking, so that it becomes easier for me to see it visually?
I appreciate your thoughts on it. Also, if any references that gradually improve my thinking on this type questions will also welcome.
Suppose you have a bilinear map $b: X\times Y\to Z$. You want to associate to it a linear map $X\to L(Y,Z)$. So for each $x\in X$, you want a linear map $f_x: Y\to Z$ (and you want that association $x\mapsto f_x$ to be linear itself).
Now how can you "naturally" define a linear map $f_x: Y\to Z$ given $b$ and $x$? There are honestly not that many possibilities: one is naturally inclined to define $f_x: y\mapsto b(x,y)$.
You should check that this association $b\mapsto (x\mapsto f_x)$ gives the isomorphism you are looking for.