I am a computer science student and do not have any background about tensors. Recently, I started learning about tensor spaces and tensor products from the text "Tensor spaces and exterior algebra" by Takeo Yokonuma.
I had a question and I understand that its pretty basic - but I would like some help with that.
The author is trying to show that given two vector spaces $V$ and $W$ over a field $k$ and any bilinear map $\phi \in L(V,W,U)$ (EDIT: where $L(V,W,U)$ is the set of all bilinear maps from $V \times W \rightarrow U$ for various vector spaces $U$ over the field $k$.) We consider a particular bilinear mapping $l \colon V \times W \to U_0$ (i) there exists a $k$-linear mapping $F \ \colon U_0 \to U$ such that $\phi = F \ o \ l$. () (ii) $l$ (in the former part) is the mapping from $V \times W$ to $U_0$.
I was going through the proof of part (i) which is where my troubles begin.
Author has already established that the $dim(U_0) \geq mn$ where $dim(V) = n$ and $dim(W) = m$. He proceeds by picking a $U_0$ of dimension exactly $mn$. Thereafter, he takes a basis $\mathcal{G} = {g_1, g_2 \ldots g_{mn}}$ of $U_0$ and observes that the basis $\mathcal{E}$ of $U$ and $\mathcal{F}$ of $W$ are such that we have a one-to-one correspondence between the sets $\mathcal{E} \times \mathcal{F}$ and $\mathcal{G}$.
After this observation, the author lets $\phi \in L(V,W,U)$ for a vector space $U$. Then, he defines $F_{\phi} \colon U_0 \to U$ by
$F_{\phi}(u) = F_{\phi}(\sum_{i,j}\gamma_{i,j} \cdot g(i,j) = \sum_{i,j} \gamma_{i,j} \cdot \phi(i,j))$
and this is what I have trouble believing. I am willing to believe that $F_{\phi} (\gamma_{ij}g(i,j)) = \gamma_{ij} \phi(i,j)$ but I cannot assume that this also distributes linearly - as this is what I want to prove.
Maybe it is easy to see but I would like to have some help here (as I do not see why is this definition for $F_{\phi}$ legitimate)
Hope my question is clear. Please help Thanks -Akash