Schauder basis

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In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis. The difference is that for Hamel bases, linear combinations are assumed to be finite sums, while for Schauder bases they may be infinite. This makes Schauder bases more suitable for the analysis of infinite-dimensional vector spaces.

A Banach space with a Schauder basis is necessarily separable, but the converse is false; that is, there exists a separable Banach space without a Schauder basis. A Banach space with a Schauder basis has the approximation property.

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[edit] Definition

Let V denote a topological vector space (for example, a Banach space or a Hilbert space) over the field F. A Schauder basis is a countable subset B of V such that every element v ∈ V can be written uniquely as a series

v = \sum_{x \in B} a_x x \,

where the infinite sum is to be understood as a limit of a sequence of finite partial sums, with ax ∈ F in exactly one way.

A Schauder basis is said to be unconditional if the series \sum_{x \in B} a_x x converges unconditionally.

[edit] Example

The archetypal example of a Schauder basis is the Fourier series of a function: every square-integrable real-valued periodic function f with period 2π can be written as a Fourier series

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

in exactly one way. The above equality is to be understood in the L2 sense: the series converges in the L2 space to f (see convergence of Fourier series for details). This proves that

\{ 1, \cos x, \sin x, \cos(2x), \sin(2x), \cos(3x), \sin(3x), \ldots \} \,

is a Schauder basis for the space of square-integrable periodic functions with period 2π.

[edit] Related concepts

A Hamel basis is a subset B of a topological vector space V such that every element v ∈ V can uniquely be written as

v = \sum_{x \in B} a_x x \,

with ax ∈ F, with the extra condition that the set

\{ x \in B \mid a_x \neq 0 \} \,

is finite.

A family of vectors is total if its linear span (the set of finite linear combinations) is dense in V. Every complete set of vectors is total, but the converse need not hold in an infinite-dimensional space.

If V is an inner product space, an orthogonal basis is a subset B such that its linear span is dense in V and elements in the basis are pairwise orthogonal.

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This article incorporates material from Countable basis on PlanetMath, which is licensed under the GFDL.