Linear forms/Product/Multilinear/Exercise

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

$\varphi _{i}\colon V_{i}\longrightarrow K$

(${}i=1,\ldots ,n$), denote linear mappings. Show that the mapping

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

is multilinear.

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