countable basis

A countable basis β of a vector space V over a field F is a countableMathworldPlanetmath subset βV with the property that every element vV can be written as an infinite series


in exactly one way (where axF). We are implicitly assuming, without further comment, that the vector space V has been given a topological structure or normed structureMathworldPlanetmath in which the above infinite sum is absolutely convergent (so that it convergesPlanetmathPlanetmath 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 continuousMathworldPlanetmathPlanetmath real-valued periodic function f on the unit circle S1=/2π can be written as a Fourier series


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