Can anyone help me showing the following:
Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an unique linear map $f:G\rightarrow H$ such that $\psi=f\circ \varphi$ then $(G, \varphi)$ is a tensor product of $E$ and $F$.
The definition I have for tensor product of vector spaces is:
Definition: Let $E, F$ and $G$ be vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. We say the pair $(G, \varphi)$ is a tensor product of $E$ and $F$ if:
(i) $\textrm{im}(\varphi)=G$.
(ii) For every $\psi:E\times F\rightarrow H$ bilinear, where $H$ is an arbitrary vector space, there is linear map $f:G\rightarrow H$ such that $\psi=f\circ \varphi$.
So using this definition it suffices showing the map $\varphi:E\times F\rightarrow G$ is surjective for solving my problem...
Any help will be welcome..Thanks