Hom/As tensor product/Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ denote a ${}K$-vector space. Let ${}{V}^{*}$ be the dual space of ${}V$. Show the following statements.

a) There exists a multilinear mapping

${V}^{*}\times W\longrightarrow \operatorname {Hom} _{K}{\left(V,W\right)},(f,w)\longmapsto {\left(v\mapsto f(v)w\right)}.$

b) There exists a linear mapping

$\psi \colon {V}^{*}\otimes W\longrightarrow \operatorname {Hom} _{K}{\left(V,W\right)}$

that sends ${}f\otimes w$ to the linear mapping ${}v\mapsto f(v)w$.

c) If ${}V$ and ${}W$ are finite-dimensional, then ${}\psi$ from part (b) is an isomorphism.

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