Vector space/Tensor product/Universal property/Fact

Universal property of the tensor product

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ denote vector spaces over ${}K$.

1. The mapping

   $\pi \colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\otimes _{K}\cdots \otimes _{K}V_{n},{\left(v_{1},\ldots ,v_{n}\right)}\longmapsto v_{1}\otimes _{}\cdots \otimes _{}v_{n},$

   is ${}K$-multilinear.

2. Let ${}W$ be another ${}K$-vector space, and let

   $\psi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

   denote a multilinear mapping. Then there exists a uniquely determined ${}K$-linear mapping

   ${\bar {\psi }}\colon V_{1}\otimes _{K}\cdots \otimes _{K}V_{n}\longrightarrow W$

   fulfilling ${}\psi ={\bar {\psi }}\circ \pi$.

Proof, Write another proof