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

v=xβaxx

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

f(x)=n=0ancos(nx)+n=1bnsin(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