Schauderbasis

In the functional analysis is a countable set of a Banach space whose linear span is dense in the whole space, called a Schauder basis if each vector with respect to her has a unique representation as a (infinite ) linear combination.

Named are the Schauder bases after the Polish mathematician Juliusz Schauder (1899-1943), who described it in 1927.

Definition

Be a Banach space over the base or. A sequence in is called Schauder basis if any can be uniquely represented as a convergent series.

Examples

• In the next room with the ℓ p-norm form a Schauder basis for the unit vectors.

• Set for all, and, by defining Up to a constant factor, each limited to [ 0,1) Haar wavelet function. As a result, the one after Alfréd hair also called the hair system is a Schauder basis for the space Lp ( [0,1] ) for.

Properties

Main Features

• A Banach space which has a Schauder basis is separable.
• Conversely, not every separable Banach space a Schauder basis.
• Banach spaces with Schauder basis have the approximation property.
• In infinite-dimensional Banach spaces Schauder basis is never a Hamel basis of the vector space, since such an infinite-dimensional Banach spaces in must be uncountable always ( see Theorem of Baire ).
• The representation of an element with respect to a Schauder basis is unique by definition. The assignments are referred to as the coefficient functionals; they are linear and continuous, and therefore elements of the dual space of.

Properties of the base

Schauder bases can have further properties. The existence of Schauder bases with such features then has further consequences for the Banach space.

Is a Schauder basis of the Banach space, then there exists a constant such that for every choice of scalars and the inequality holds. The infimum over those who meet predetermined base this inequality is called the basis constant. One speaks of a monotone basis if the base constant is equal to 1.

This is called a base limited completely (English: boundedly complete) if there is to be any sequence of scalars with a with.

Next is the closed subspace generated by, and is the norm of the restricted functional. The base is called shrinking (English: shrinking ) if for all.

Finally, one speaks of an unconditional basis (English: unconditional ) if all rows in the developments concerning the base converge absolutely. It can be shown that the hair - system is for an absolute basis, but not. The room has no unconditional basis.

Two sets of R. C. James

The following two sets of Robert C. James demonstrate the importance of basic terms.

• RC James: Let be a Banach space with a Schauder basis. is reflexive if the base is completely limited and shrinking.

For unconditional Schauder bases can characterize the presence of certain subspaces. Be a Banach space with an unconditional Schauder basis. Then:

• Contains no isomorphic to c0 subspace. The base is completely restricted.
• Contains no isomorphic to the subspace. The base is shrinking.

As a consequence, in these scenarios:

• RC James: Let be a Banach space with an unconditional Schauder basis. is reflexive, if not to or isomorphic subspace contains.