# countable basis

A countable basis $\beta $ of a vector space $V$ over a field $F$ is a countable^{} subset $\beta \subset V$ with the property that every element $v\in V$ can be written as an infinite series

$$v=\sum _{x\in \beta}{a}_{x}x$$ |

in exactly one way (where ${a}_{x}\in F$). We are implicitly assuming, without further comment, that the vector space $V$ has been given a topological structure or normed structure^{} in which the above infinite sum is absolutely convergent (so that it converges^{} to $v$ regardless of the order in which the terms are summed).

The archetypical example of a countable basis is the Fourier series of a function: every continuous^{} real-valued periodic function $f$ on the unit circle ${S}^{1}=\mathbb{R}/2\pi $ can be written as a Fourier series

$$f(x)=\sum _{n=0}^{\mathrm{\infty}}{a}_{n}\mathrm{cos}(nx)+\sum _{n=1}^{\mathrm{\infty}}{b}_{n}\mathrm{sin}(nx)$$ |

in exactly one way.

Note: A countable basis is a countable set, but it is not usually a basis.

Title | countable basis |
---|---|

Canonical name | CountableBasis |

Date of creation | 2013-03-22 12:10:37 |

Last modified on | 2013-03-22 12:10:37 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 42-00 |

Classification | msc 15A03 |

Synonym | Schauder basis |