PlanetMath: countable basis

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countable basis

(Definition)

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

$\displaystyle v = \sum_{x \in \beta} a_x x$

in exactly one way (where $a_x \in F$ ). We are implicitly assuming, without further comment, that the vector space

has been given a topological structure or normed structure in which the above infinite sum is absolutely convergent (so that it converges to

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 on the unit circle $S^1 = \mathbb{R}/2\pi$ can be written as a Fourier series

$\displaystyle f(x) = \sum_{n=0}^\infty a_n \cos(n x) + \sum_{n=1}^\infty b_n \sin(n x)$

in exactly one way.

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

"countable basis" is owned by djao.

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Other names:

Schauder basis

Attachments:: every normed space with Schauder basis is separable (Theorem) by asteroid
example of construction of a Schauder basis (Example) by perucho

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AMS MSC:	42-00 (Fourier analysis :: General reference works )
	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

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