Contents

Definition

A for $k$ a field, a vector space over $k$ is module over the ring $k$. Sometimes a vector space over $k$ is called a $k$-linear space. (Compare ‘$k$-linear map’.)

The category of vector spaces is typically denoted Vect, or ${\mathrm{Vect}}_k$ if we wish to make the field $k$ explicit. So

This category has vector spaces over $k$ as objects, and $k$-linear maps between these as morphisms.

Properties

By the basis theorem (and using the axiom of choice) every vector space admits a basis.

Related concepts

• vector bundle,

• lattice in a vector space

• vector space, dual vector space,

  • topological vector space, convenient vector space

• 2-vector space

• n-vector space