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.