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countable basis (Definition)

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

$\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 $ 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

$\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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Cross-references: basis, unit circle, periodic function, continuous, function, Fourier series, terms, order, converges, absolutely convergent, sum, structure, series, infinite, property, subset, countable, field, vector space
There are 8 references to this entry.

This is version 4 of countable basis, born on 2002-01-07, modified 2002-06-20.
Object id is 1434, canonical name is CountableBasis.
Accessed 4834 times total.

Classification:
AMS MSC42-00 (Fourier analysis :: General reference works )
 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

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