nLab
basis of a vector space

Contents

Definition

For kk a field and VV a kk-vector space, a basis for VV is a basis of a free module for VV regarded as a free module over kk. In functional analysis, a basis in this sense is called a Hamel basis.

Properties

The basis theorem asserts that, with the axiom of choice, every vector space admits a basis, hence that every module over a field is a free module.

Examples

In representation theory:

Specifically in representation theory of the symmetric group: