Linear mappings/Tensor product/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}}$ denote vector spaces over ${\displaystyle {}K}$. For ${\displaystyle {}K}$-linear mappings

${\displaystyle \varphi _{i}\colon V_{i}\longrightarrow W_{i},}$

the linear mapping

${\displaystyle V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow W_{1}\otimes _{}\cdots \otimes _{}W_{n},v_{1}\otimes _{}\cdots \otimes _{}v_{n}\longmapsto \varphi _{1}(v_{1})\otimes _{}\cdots \otimes _{}\varphi _{n}(v_{n}),}$

is called the tensor product of the ${\displaystyle {}\varphi _{i}}$. It is denoted by ${\displaystyle {}\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n}}$.