I am reading "Riemannian Geometry" by Gallot. And I am confused with the following definition of tensor product:
Let $E$ and $F$ are two finite dimensional vector spaces, a vector space $E\otimes F$, unique up to isomorphism and such that for any vector space $G$, $L(E\otimes F,G)$ is isomorphic to $L_2(E\times F,G)$ (the vector space of bilinear maps from $E\times F$ to $G$): $E\otimes F$ is the tensor product of $E$ and $F$. Moreover, there exists a bilinear map from $E\times F$ to $E\otimes F$, denoted by $\otimes$, and such that if $e_i$ and $f_j$ are basis for $E$ and $F$, $(e_i\otimes f_j)$ is basis for $E\otimes F$.
I am confused about the words after "moreover". It seems that the words before "moreover" is the definition of tensor product of two vector spaces. But I don't know how to prove the existence of the bilinear map satisfying the basis requirement. It seems that the definition of tensor product on some online sources also includes the existence of the bilinear map. Which one is correct? This is the first time I encounter tensor product. Thanks!