However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis.

The space ℓ is not separable, and therefore has no Schauder basis.

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions.

Banach asked whether every separable Banach space have a Schauder basis.

Schauder basis (in a Banach space)

Every orthonormal basis in a separable Hilbert space is a Schauder basis.

The Franklin system provides a Schauder basis in the disk algebra A(D).

A Banach space with a Schauder basis is necessarily separable, but the converse is false, as described below.

In 1972, Per Enflo constructed a separable Banach space that lacks the approximation property and a Schauder basis.

Can any function f be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise?

This was negatively answered by Per Enflo who constructed a separable Banach space failing the approximation property, thus a space without a Schauder basis.

The most important alternatives are orthogonal basis on Hilbert spaces, Schauder basis and Markushevich basis on normed linear spaces.

In a long monograph, Grothendieck proved that if every Banach space had the approximation property, then every Banach space would have a Schauder basis.

The first example by Enflo of a space failing the approximation property was at the same time the first example of a Banach space without Schauder basis.

With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite-dimensional subspace that admits an unconditional Schauder basis.

Every space with a Schauder basis has the AP (we can use the projections associated to the base as the 's in the definition), thus a lot of spaces with the AP can be found.

Robert C. James characterized reflexivity in Banach spaces with basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.

In functional analysis and related areas of mathematics an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis.

This condition is weaker than the existence of the Schauder basis, but formally stronger than the classical approximation property (however, it is not clear (2012) whether the stereotype approximation property coincide with the classical one, or not).

Bočkarev's construction of a Schauder basis in A(D) goes as follows: let f be a complex valued Lipschitz function on [0, π]; then f is the sum of a cosine series with absolutely summable coefficients.

Since every vector v in a Banach space V with a Schauder basis is the limit of P(v), with P of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property.

In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums.

On 6 November 1936, he posed the "basis problem" of determining whether every Banach space has a Schauder basis, with Mazur promising a "live goose" as a reward: Thirty-seven years later, a live goose was awarded by Mazur to Per Enflo in a ceremony that was broadcast throughout Poland.