Tensor product/Mapping spaces/Product set/Exercise

Let ${}K$ be a field, and let ${}I$ and ${}J$ denote finite index sets. Show that the mapping

$\operatorname {Map} \,{\left(I,K\right)}\times \operatorname {Map} \,{\left(J,K\right)}\longrightarrow \operatorname {Map} \,{\left(I\times J,K\right)},(f,g)\longmapsto f\otimes g,$

given by

${}(f\otimes g)(i,j):=f(i)\cdot g(j)\,,$

is multilinear.

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