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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix $\text B$. Review of Tensors