Consider the standard tensor algebra over a vector space: $$T(V)=\bigoplus_{n\in N}V^{\otimes n}$$, I am trying to define a linear map from $T(V)\times T(V)\rightarrow T(V)$. Up to this time, I have a isomorphism from $V^{\otimes n}\otimes V^{\otimes m}\rightarrow V^{\otimes (n+m)}$ and trivially bilinear map from $V^{\otimes n}\times V^{\otimes m}\rightarrow V^{\otimes (n+m)}$.
Now,consider an element $v=v_{i_1}+v_{i_2}+...+v_{i_n},w=w_{i_1}+w_{i_2}+...+w_{i_m}$ in $T(V)$, I could define $(v,w)\rightarrow \sum_{n,m}v_{i_n}w_{i_m}$, but I can't see why this is linear and not bilinear.
How do I define a linear multiplication from $T(V)\times T(V)\rightarrow T(V)$?