# Definition:Tensor

## Definition

Let $V$ and $V^*$ be a vector space and its dual.

Then a (mixed) tensor $F$ of type $\tuple {k, l}$ is a multilinear map such that:

$\ds F : \underbrace{V \times \ldots \times V}_{\text{$k$times}} \times \underbrace{{V^*} \times \ldots \times {V^*}}_{\text{$l$times}} \to \R$

## Also known as

A tensor of type $\tuple {k, l}$ is also known as a $k$-times covariant and $l$-times contravariant tensor.