Let $E, F$ be vector spaces and $G$ be a normed space. Let $B(E\times F,G)$ the set of all bilinear forms on $E \times F$ and $L(E \otimes F,G)$ be the set of all linear mappings from $E \otimes F$ to $G.$ I need to
show that $$B(E \times F,G)=L(E \otimes F,G).$$
My question is what is meant by equality here?
Let $f \in B(E \times F,G).$ Define $g:E \otimes F \to G$ as $$g(x \otimes y)=f(x,y)$$ and extend linearly. Then $g \in L(E \otimes F,G).$
Define $T:B(E \times F,G)\to L(E \otimes F,G)$ as $Tf=g.$ I have shown that $T$ is a isometry.
What else do I need to show?