Vector space/Tensor product/Dual space/Relation/Fact

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

Then there exists a natural

isomorphism

${V_{1}}^{*}\otimes _{}\cdots \otimes _{}{V_{n}}^{*}\longrightarrow {\left(V_{1}\otimes _{}\cdots \otimes _{}V_{n}\right)}^{*},f_{1}\otimes _{}\cdots \otimes _{}f_{n}\longmapsto {\left(v_{1}\otimes _{}\cdots \otimes _{}v_{n}\mapsto f_{1}(v_{1})\cdots f_{n}(v_{n})\right)}.$

Proof, Write another proof