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Suppose that we defined a tensor product of vector spaces $U$ and $V$ as a quotient of a vector space with basis $V \times W$ by the vector space spanned by $-(u_1+u_2,v)+(u_1,v)+(u_2,v), -(u,v_1+v_2)+(u,v_1)+(u,v_2),-(au,v)+(u,av)$.

I have proved the universal property for two vector spaces. But how then extend it to a tensor product of $n$ vector spaces?

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What goes wrong with trying to do the same thing? –  Hurkyl Dec 2 '13 at 9:59

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You can define it recursively by $V_1 \otimes \cdots \otimes V_n = \left( V_1 \otimes \cdots \otimes V_{n-1} \right) \otimes V_n$, but you want to check associativity.

See this answer.

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Well, I proved that for vector spaces $U,V,W$, $U \otimes (V \otimes W) \cong (U \otimes V) \otimes W$. But how can we now "recursively" say that for any multilinear map $V_1 \times ... \times V_n \rightarrow U$ we have a unique linear transformation $V_1 \otimes ... \otimes V_n \rightarrow U$? –  user41468 Dec 2 '13 at 10:23

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