I need some clarification on the idea behind tensor product. I've read a lot about this topic in different sources, but still have big mess in my head.
Let's start from the multilinear maps - a definition of tensors which I believe I do understand. For a given vector space $V$, with base $\{e_\mu\}$ and $\dim V=n$, all possible linear maps $\varphi_v(v):V\rightarrow\mathbb{R}$, a so called covectors, form themself a vector space $V^*$, called dual with basis $\{e^\mu\}$. Similarly space $W$ ($\dim W=m$, base $\{f_\nu\}$) has its $W^*$ spanned by $\varphi_w$ with basis $\{f^\nu\}$.
Now, we can create a Cartesian product $V\times W$, with dual $(V\times W)^*$ formed by bilinear maps $\varrho(v,w): V\times W\rightarrow\mathbb{R}$, and the problem arises: what is the relation between its covectors and individual covectors of $V$ and $W$. To address this issue we create yet another vector space, the tensor product space, which consists of all linear combinations of initial covectors of the form $T_{\mu\nu} e^\mu\otimes f^\nu$. The $\dim V^*\otimes W^*=nm$ is obvious because it is "everybody with everybody". So far, so good. Now, if I understand correctly, they claim that if we act with vector from tensor product space on vector from Cartesian product space, the bilinearity operator factorize, i.e. $$[T_{\mu\nu} e^\mu\otimes f^\nu](\alpha^\gamma e_\gamma,\beta^\lambda f_\lambda)=T_{\mu\nu}\alpha^\gamma\beta^\lambda e^\mu(e_\gamma)f^\nu(f_\lambda).$$ I cannot understand where did it come from, but some authors just start from this like its obvious. I've used a word factorization, because for me it looks that in this picture $\varrho(v,w)=\varphi_v(v)\varphi_w(w)$. Perhaps I completely misunderstood the idea, can anyone shed some light on this, please?