Editing Banach space (section)

==Schauder bases==
{{main|Schauder basis}}

A ''Schauder basis'' in a Banach space <math>X</math> is a sequence <math>\{e_n\}_{n \geq 0}</math> of vectors in <math>X</math> with the property that for every vector <math>x \in X,</math> there exist {{em|uniquely}} defined scalars <math>\{x_n\}_{n \geq 0}</math> depending on <math>x,</math> such that
<math display=block>x = \sum_{n=0}^{\infty} x_n e_n, \quad \textit{i.e.,} \quad x = \lim_n P_n(x), \ P_n(x) := \sum_{k=0}^n x_k e_k.</math>

Banach spaces with a Schauder basis are necessarily [[Separable space|separable]], because the countable set of finite linear combinations with rational coefficients (say) is dense.

It follows from the Banach–Steinhaus theorem that the linear mappings <math>\{P_n\}</math> are uniformly bounded by some constant <math>C.</math> 
Let <math>\{e_n^*\}</math> denote the coordinate functionals which assign to every <math>x</math> in <math>X</math> the coordinate <math>x_n</math> of <math>x</math> in the above expansion. 
They are called ''biorthogonal functionals''. When the basis vectors have norm <math>1,</math> the coordinate functionals <math>\{e_n^*\}</math> have norm <math>{}\leq 2 C</math> in the dual of <math>X.</math>

Most classical separable spaces have explicit bases. 
The [[Haar wavelet|Haar system]] <math>\{h_n\}</math> is a basis for <math>L^p([0, 1])</math> when <math>1 \leq p < \infty.</math>  
The [[Schauder basis#Examples|trigonometric system]] is a basis in <math>L^p(\mathbf{T})</math> when <math>1 < p < \infty.</math> 
The [[Haar wavelet#Haar system on the unit interval and related systems|Schauder system]] is a basis in the space <math>C([0, 1]).</math><ref>see {{harvtxt|Lindenstrauss|Tzafriri|1977}} p.&nbsp;3.</ref> 
The question of whether the disk algebra <math>A(\mathbf{D})</math> has a basis<ref>the question appears p.&nbsp;238, §3 in Banach's book, {{harvtxt|Banach|1932}}.</ref> remained open for more than forty years, until Bočkarev showed in 1974 that <math>A(\mathbf{D})</math> admits a basis constructed from the [[Haar wavelet#Haar system on the unit interval and related systems|Franklin system]].<ref>see S. V. Bočkarev, "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system". (Russian) Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159.</ref>

Since every vector <math>x</math> in a Banach space <math>X</math> with a basis is the limit of <math>P_n(x),</math> with <math>P_n</math> of finite rank and uniformly bounded, the space <math>X</math> satisfies the [[Approximation property|bounded approximation property]]. 
The first example by [[Per Enflo|Enflo]] of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.<ref>see {{cite journal|last1=Enflo|first1=P.|year=1973|title=A counterexample to the approximation property in Banach spaces|journal=Acta Math.|volume=130|pages=309–317| doi=10.1007/bf02392270| s2cid=120530273 | doi-access=free}}</ref>

Robert C. James characterized reflexivity in Banach spaces with a basis: the space <math>X</math> with a Schauder basis is reflexive if and only if the basis is both [[Schauder basis#Schauder bases and duality|shrinking and boundedly complete]].<ref>see R.C. James, "Bases and reflexivity of Banach spaces". Ann. of Math. (2) 52, (1950). 518–527. See also {{harvtxt|Lindenstrauss|Tzafriri|1977}} p.&nbsp;9.</ref> 
In this case, the biorthogonal functionals form a basis of the dual of <math>X.</math>