Tensor products

I'm trying to get my head round tensor products of vector spaces (I'm happy to see arguments in a more general setting, though).

I am concerned principally with two statements:

i) If $U,V,W$ are vector spaces then there is a one-to-one correspondence $\{ \mathrm{linear \ maps} \ V \otimes W \to U \} \longleftrightarrow \{ \mathrm{bilinear \ maps} \ V\times W \to U \}$.

ii) There is a natural (basis-independent) isomorphism $(U \oplus V) \otimes W \to (U \otimes W) \oplus (V \otimes W)$

For the first of these statements, I can see map from left to right; any linear map $\phi : V \otimes W \to U$ gives rise to a bilinear map $V \times W \to V \otimes W \to U$, where the first of these maps is the canonical map $p: (v,w) \mapsto v \otimes w$ and the second is $\phi$. I can't see, however, why any bilinear map $V \times W \to U$ necessarily factors into $\phi \circ p$ for some suitable linear map $\phi$.

I haven't got much experience with commutative diagrams. I think I've convinced myself that ii) is true with a commutative diagram, but I don't know if it's correct (and I also don't know how to LaTeX it easily...)

Any help would be appreciated. Thanks!

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i) is usually part of the definition of a tensor product. What definition are you working with? – Qiaochu Yuan Mar 22 '12 at 19:00
@QIaochu: it sounds like Matt has a construction of tensor products anbd is trying to prove it has the correct properties. – Mariano Suárez-Alvarez♦ Mar 22 '12 at 19:05
@QiaochuYuan The definition I have is that if $V,W$ are vector spaces (with bases $v_1, \ldots , v_m$ and $w_1, \ldots w_n$), then the tensor product of $V$ and $W$ is the vector space with basis $\{ v_i \otimes w_j \ | \ 1 \leq i \leq m, 1 \leq j \leq n \}$. My notes then go on to define the tensor product of $v \in V$, $w \in W$ to be $v \otimes w = \sum \lambda_i v_i \otimes \sum \mu_j w_j = \sum_{i,j} \lambda_i \mu_j (v_i \otimes w_j)$ – Matt Mar 22 '12 at 19:33
This definition does seem strange to me, since it doesn't come with any intuition (it just seems like formal sums of symbols, which I don't like) – Matt Mar 22 '12 at 19:35

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– KCd Mar 22 '12 at 20:46
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1 Answer

In your definition (you should really look up the universal property of the tensorproduct!) you can argue as follows:

If $b$ is a bilinear map $V\times W \rightarrow U$, you can simply define a linear map $$v_i \otimes w_j \mapsto b(v_i,w_j)$$ since you know the $v_i\otimes w_j$ are a basis and a linear map can be defined by choosing arbitrary values for the elements of a basis.

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