#
nLab

basis of a vector space

# Contents

## Definition

For $k$ a field and $V$ a $k$-vector space, a **basis** for $V$ is a basis of a free module for $V$ regarded as a free module over $k$. 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: