$X, Y$ are vector spaces. A vector space $W$ and a map $A:X\times Y\rightarrow W$ satisfy the following:
for every bilinear map T from $X\times Y$ to a vector space $Z$, there exists a unique linear mapping $L:W\rightarrow Z$ such that $L\circ A=T$. (In fact this is the definition of tensor product)
Then how can I show the image of bilinear mapping $A$ spans $W$?