nLab
vector space

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

$${\mathrm{Vect}}_k≔k\mathrm{Mod}.$$Vect_k \coloneqq k Mod \,.

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

Related concepts

• vector space, dual vector space,

  • topological vector space, convenient vector space

• 2-vector space

• n-vector space