# Universality of tensor product

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)

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What is $U_0$? Having undefined things pop out from nowhere is bad form. I think you mean $U_0 = V \otimes W$? – kahen Jan 15 '11 at 8:48
I second that.... – user02138 Jan 15 '11 at 9:10
I am sorry, I will make that edit – Akash Kumar Jan 15 '11 at 13:43

Linearity is built into the definition: you first define the map $F_\phi$ on the basis elements, then you extend linearly. I am not sure what you mean by "this is what I have trouble believing". Clearly, if you only define a map on a basis and then extend linearly to linear combinations of the basis elements, you get a linear map. The only thing that you might doubt is that this is well-defined. But in the context of vector spaces there is no such issue, since every element can be uniquely written as a linear combination of the basis elements.
Thanks Alex. Let me embarrass myself a little more. I am okay with $F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0)) = \gamma_{i_0,j_0}\phi(i_0,j_0)$ and $F_{\phi}(\gamma_{i_1,j_1}g(i_1,j_1)) = \gamma_{i_1,j_1}\phi(i_1,j_1)$ but I am uncomfortable with $F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0) + \gamma_{i_1,j_1}g(i_1,j_1)) = F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0))+ F_{\phi}(\gamma_{i_1,j_1}g(i_1,j_1))$. It seems that I am assuming that $F_{\phi}$ distributes linearly over the basis elements and I am not sure why is this correct. – Akash Kumar Jan 15 '11 at 14:10
@Akash You are not assuming it, you are defining $F_\phi$ to distribute linearly. That's how you define it on a sum of elements: to be the sum of images under $F_\phi$ of each of the elements. – Alex B. Jan 15 '11 at 14:22