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 fit for the analysis of infinite-dimensional vector spaces.

1 Definition
2 Example
3 Related concepts
4 References

[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 a_x ∈ F in exactly one way.

[edit] Example

The archetypical 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 a_x ∈ 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.

An orthogonal basis is subset B such that its linear span (the set of finite linear combinations) is dense in V and elements in the basis are pairwise orthogonal. The latter condition requires V to be an inner product space.