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In [[mathematics]], a '''Schauder basis''' or '''countable basis''' is similar to the usual ([[Hamel basis|Hamel]]) [[basis (linear algebra)|basis]] of a [[vector space]]; the difference is that Hamel bases use [[linear combination]]s that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional [[topological vector space]]s including [[Banach space]]s.

Schauder bases were described by [[Juliusz Schauder]] in 1927,<ref>see {{harvtxt|Schauder|1927}}.</ref><ref name="SchauderB">{{cite journal | last1 = Schauder | first1 = Juliusz | year = 1928 | title = Eine Eigenschaft des Haarschen Orthogonalsystems | url = | journal = Mathematische Zeitschrift | volume = 28 | issue = | pages = 317–320 | doi=10.1007/bf01181164}}</ref> although such bases were discussed earlier. For example, the [[Haar basis]] was given in 1909, and [[Georg Faber|G. Faber]] discussed in 1910 a basis for continuous functions on an interval, sometimes called a '''Faber–Schauder system'''.<ref name="Faber">Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) '''19''': 104&ndash;112. {{issn|0012-0456}}; 
http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553</ref>

== Definitions ==
Let ''V'' denote a [[Banach space]] over the [[field (mathematics)|field]]&nbsp;''F''. A '''Schauder basis''' is a [[sequence]] {''b''<sub>''n''</sub>} of elements of&nbsp;''V'' such that for every element {{nowrap|''v'' ∈ ''V''}} there exists a ''unique'' sequence {α<sub>''n''</sub>} of scalars in&nbsp;''F'' so that

:<math> v = \sum_{n=0}^\infty \alpha_n b_n,</math>

where the convergence is understood with respect to the norm topology, ''i.e.'',

:<math> \lim_{n \to \infty} \left \| v - \sum_{k=0}^n \alpha_k b_k \right\|_V = 0.</math>

Schauder bases can also be defined analogously in a general [[topological vector space]]. As opposed to a [[Hamel basis]], the elements of the basis must be ordered since the series may not converge [[unconditional convergence|unconditionally]].

A Schauder basis {''b''<sub>''n''</sub>}{{nowrap|<sub>&thinsp;''n''&thinsp;&ge;&thinsp;0</sub>}} is said to be '''normalized''' when all the basis vectors have norm&nbsp;1 in the Banach space&nbsp;''V''.

A sequence {''x''<sub>''n''</sub>}{{nowrap|<sub>&thinsp;''n''&thinsp;&ge;&thinsp;0</sub>}} in ''V'' is a '''basic sequence''' if it is a Schauder basis of its [[Linear span#Closed linear span|closed linear span]].

Two Schauder bases, {''b''<sub>''n''</sub>} in ''V'' and {''c''<sub>''n''</sub>} in ''W'', are said to be '''equivalent''' if there exist two constants {{nowrap|''c'' > 0}} and ''C'' such that for every integer {{nowrap|''N'' &ge; 0}} and all sequences {α<sub>''n''</sub>} of scalars,

:<math> c \left \| \sum_{k=0}^N \alpha_k b_k \right\|_V  \le \left \| \sum_{k=0}^N \alpha_k c_k \right \|_W \le C \left \| \sum_{k=0}^N \alpha_k b_k \right \|_V.</math>

A family of vectors in ''V'' is '''total''' if its [[linear span]] (the set of finite linear combinations) is  [[dense subset|dense]] in ''V''.  If ''V'' is a [[Hilbert space]], an '''[[orthogonal basis]]''' is a ''total'' subset ''B'' of ''V'' such that elements in ''B'' are nonzero and pairwise orthogonal. Further, when each element in ''B'' has norm&nbsp;1, then ''B'' is an '''[[orthonormal basis]]''' of ''V''.

== Properties ==
Let {''b<sub>n</sub>''} be a Schauder basis of a Banach space ''V'' over '''F'''&nbsp;= '''R''' or&nbsp;'''C'''.  It is a subtle consequence of the [[Open mapping theorem (functional analysis)|open mapping theorem]] that the linear mappings {''P<sub>n</sub>''} defined by

:<math> v = \sum_{k=0}^\infty \alpha_k b_k \ \ \overset{\textstyle P_n}{\longrightarrow} \ \ P_n(v) = \sum_{k = 0}^n \alpha_k b_k</math>

are uniformly bounded by some constant ''C''.<ref>see Theorem 4.10 in {{harvtxt|Fabian|Habala|Hájek|Montesinos|2011}}.</ref> When {{nowrap|''C'' {{=}} 1}}, the basis is called a '''monotone''' basis. The maps {''P<sub>n</sub>''} are the '''[[Projection (linear algebra)|basis projections]]'''.

Let {''b*<sub>n</sub>''} denote the '''coordinate functionals''', where ''b*<sub>n</sub>'' assigns to every vector ''v'' in ''V'' the coordinate α<sub>''n''</sub> of ''v'' in the above expansion. Each ''b*<sub>n</sub>'' is a bounded linear functional on ''V''. Indeed, for every vector ''v'' in ''V'',

:<math> |b^*_n(v)| \; \|b_n\|_V = |\alpha_n| \; \|b_n\|_V = \|\alpha_n b_n\|_V = \|P_n(v) - P_{n-1}(v)\|_V \le 2 C \|v\|_V.</math>

These functionals {''b*<sub>n</sub>''} are called '''biorthogonal functionals''' associated to the basis {''b''<sub>''n''</sub>}. When the basis {''b''<sub>''n''</sub>} is normalized, the coordinate functionals {''b*<sub>n</sub>''} have norm &le; 2''C'' in the [[Dual space#Continuous dual space|continuous dual]] {{nowrap|''V''&thinsp;′}} of&nbsp;''V''.

A Banach space with a Schauder basis is necessarily [[Separable space|separable]], but the converse is false. Since every vector ''v'' in a Banach space ''V'' with a Schauder basis is the limit of ''P<sub>n</sub>''(''v''), with ''P<sub>n</sub>'' of finite rank and uniformly bounded, such a space ''V'' satisfies the [[approximation property|bounded approximation property]].

A theorem attributed to [[Stanisław Mazur|Mazur]]<ref>for an early published proof, see p.&nbsp;157, C.3 in Bessaga, C. and Pełczyński, A. (1958), "On bases and unconditional convergence of series in Banach spaces", Studia Math. '''17''': 151&ndash;164. In the first lines of this article, Bessaga and Pełczyński write that Mazur's result appears without proof in Banach's book &mdash;to be precise, on p.&nbsp;238&mdash; but they do not provide a reference containing a proof.</ref> asserts that every infinite-dimensional Banach space ''V'' contains a basic sequence, ''i.e.'', there is an infinite-dimensional subspace of ''V'' that has a Schauder basis. The '''basis problem''' is the question asked by Banach, whether every separable Banach space has a Schauder basis. This was negatively answered by [[Per Enflo]] who constructed a separable Banach space failing the approximation property, thus a space without a Schauder basis.<ref>{{Cite journal| last=Enflo | first=Per | authorlink=Per Enflo | title= A counterexample to the approximation problem in Banach spaces | journal=Acta Mathematica | volume=130 | issue=1 |date = July 1973| pages=309–317 | doi=10.1007/BF02392270 }}</ref>

== Examples ==
The standard unit vector bases of [[sequence space|''c''<sub>0</sub>]], and of [[sequence space|ℓ<sup>''p''</sup>]] for 1&nbsp;≤ ''p''&nbsp;< ∞, are monotone Schauder bases. In this '''unit vector basis''' {''b<sub>n</sub>''}, the vector ''b<sub>n</sub>'' in {{nowrap|''V'' {{=}} ''c''<sub>0</sub>}} or in {{nowrap|''V'' {{=}} ℓ<sup>''p''</sup>}} is the scalar sequence {{nowrap|&#123;''b''<sub>&thinsp;''n'',&thinsp;''j''&thinsp;</sub>&#125;<sub>&thinsp;''j''</sub>}} where all coordinates ''b<sub>n, j</sub>'' are 0, except the ''n''th coordinate:
:<math>b_n = \{b_{n, j}\}_{j=0}^\infty \in V, \ \ b_{n, j} = \delta_{n, j},</math>
where &delta;<sub>''n, j''</sub> is the [[Kronecker delta]]. The space ℓ<sup>∞</sup> is not separable, and therefore has no Schauder basis.

Every [[orthonormal basis]] in a separable [[Hilbert space]] is a Schauder basis. Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ<sup>2</sup>.

The [[Haar wavelet|Haar system]] is an example of a basis for [[Lp space|''L''<sup>''p''</sup>([0,&nbsp;1])]], when 1&nbsp;≤ ''p''&nbsp;< ∞.<ref name="SchauderB" />
When {{nowrap|1&nbsp;&lt; ''p''&nbsp;&lt; ∞}}, another example is the trigonometric system defined below. The Banach space ''C''([0,&nbsp;1]) of continuous functions on the interval [0,&nbsp;1], with the [[supremum norm]], admits a Schauder basis. The [[Haar wavelet#Haar system on the unit interval and related systems|Faber&ndash;Schauder system]] is the most commonly used Schauder basis for&nbsp;''C''([0,&nbsp;1]).<ref name="Faber" /><ref>see pp.&nbsp;48&ndash;49 in {{harvtxt|Schauder|1927}}. Schauder defines there a general model for this system, of which the Faber&ndash;Schauder system used today is a special case.</ref>

Several bases for classical spaces were discovered before Banach's book appeared ({{harvtxt|Banach|1932}}), but some other cases remained open for a long time. For example, the question of whether the [[disk algebra]] ''A''(''D'') has a Schauder basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the [[Haar wavelet#Haar system on the unit interval and related systems|Franklin system]] exists in&nbsp;''A''(''D'').<ref>see Bočkarev, S. V. (1974), "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system", (in Russian) ''Mat. Sb''. (N.S.) '''95'''(137): 3&ndash;18, 159. Translated in Math. USSR-Sb. '''24''' (1974), 1–16. The question is in Banach's book, {{harvtxt|Banach|1932}} p.&nbsp;238, &sect;3.</ref> One can also prove that the periodic Franklin system<ref>See p.&nbsp;161, III.D.20 in {{harvtxt|Wojtaszczyk|1991}}.</ref>  is a basis for a Banach space ''A''<sub>''r''</sub> isomorphic to ''A''(''D'').<ref>See p.&nbsp;192, III.E.17 in {{harvtxt|Wojtaszczyk|1991}}.</ref>
This space ''A''<sub>''r''</sub> consists of all complex continuous functions on the unit circle '''T''' whose [[Harmonic conjugate|conjugate function]] is also continuous. The Franklin system is another Schauder basis for ''C''([0,&nbsp;1]),<ref>{{cite journal | last1 = Franklin | first1 = Philip | year = 1928 | title = A set of continuous orthogonal functions | url = | journal = Math. Ann. | volume = 100 | issue = | pages = 522–529 | doi=10.1007/bf01448860}}</ref> 
and it is a Schauder basis in ''L''<sup>''p''</sup>([0,&nbsp;1]) when {{nowrap|1 &le; ''p'' &lt; &infin;}}.<ref>see p.&nbsp;164, III.D.26 in {{harvtxt|Wojtaszczyk|1991}}.</ref> 
Systems derived from the Franklin system give bases in the space ''C''<sup>1</sup>([0,&nbsp;1]<sup>2</sup>) of differentiable functions on the unit square.<ref>{{cite journal | last1 = see Ciesielski | first1 = Z | year = 1969 | title = A construction of basis in ''C''<sup>1</sup>(''I''<sup>2</sup>) | url = | journal = Studia Math. | volume = 33 | issue = | pages = 243–247 }} and {{cite journal | last1 = Schonefeld | first1 = Steven | year = 1969 | title = Schauder bases in spaces of differentiable functions | url = | journal = Bull. Amer. Math. Soc. | volume = 75 | issue = | pages = 586–590 | doi=10.1090/s0002-9904-1969-12249-4}}</ref> The existence of a Schauder basis in ''C''<sup>1</sup>([0,&nbsp;1]<sup>2</sup>) was a question from Banach's book.<ref>see p.&nbsp;238, &sect;3 in {{harvtxt|Banach|1932}}.</ref>

=== Relation to Fourier series ===
Let {''x''<sub>''n''</sub>} be, in the real case, the sequence of functions

:<math> \{ 1, \cos (x), \sin (x), \cos(2x), \sin(2x), \cos(3x), \sin(3x), \ldots \}</math>

or, in the complex case,

:<math> \left \{ 1, e^{ix}, e^{-ix}, e^{2ix}, e^{-2ix}, e^{3ix}, e^{-3ix}, \ldots \right \}.</math>

The sequence {''x''<sub>''n''</sub>} is called the '''trigonometric system'''. It is a Schauder basis for the space [[Lp space|''L''<sup>''p''</sup>([0,&nbsp;2''π''])]] for any ''p'' such that {{nowrap|1 &lt; ''p'' &lt; ∞}}. For ''p''&nbsp;=&nbsp;2, this is the content of the [[Riesz–Fischer theorem]], and for ''p''&nbsp;&ne;&nbsp;2, it is a consequence of the boundedness on the space ''L''<sup>''p''</sup>([0,&nbsp;2''π'']) of the [[Hilbert transform#Hilbert transform on the circle|Hilbert transform on the circle]]. It follows from this boundedness that the projections ''P''<sub>''N''</sub> defined by

:<math> \left \{ f : x \to \sum_{k=-\infty}^{+\infty} c_k e^{i k x} \right \} \ \overset{P_N}{\longrightarrow} \ \left \{ P_N f : x \to \sum_{k=-N}^{N} c_k e^{i k x} \right \}</math>

are uniformly bounded on ''L''<sup>''p''</sup>([0,&nbsp;2''π'']) when {{nowrap|1 &lt; ''p'' &lt; ∞}}. This family of maps {''P''<sub>''N''</sub>} is [[Equicontinuity|equicontinuous]] and tends to the identity on the dense subset consisting of [[trigonometric polynomial]]s. It follows that ''P''<sub>''N''</sub>&thinsp;''f'' tends to ''f'' in ''L''<sup>''p''</sup>-norm for every {{nowrap| ''f'' &isin; ''L''<sup>''p''</sup>([0,&nbsp;2''π''])}}. In other words, {''x''<sub>''n''</sub>} is a Schauder basis of ''L''<sup>''p''</sup>([0,&nbsp;2''π'']).<ref>see p.&nbsp;40, II.B.11 in {{harvtxt|Wojtaszczyk|1991}}.</ref>

However, the set {''x<sub>n</sub>''} is not a Schauder basis for ''L''<sup>1</sup>([0,&nbsp;2''π'']). This means that there are functions in ''L''<sup>1</sup> whose Fourier series does not converge in the ''L''<sup>1</sup> norm, or equivalently, that the projections ''P''<sub>''N''</sub> are not uniformly bounded in ''L''<sup>1</sup>-norm. Also, the set {''x<sub>n</sub>''} is not a Schauder basis for ''C''([0,&nbsp;2''π'']).

=== Bases for spaces of operators ===
The space ''K''(ℓ<sup>2</sup>) of [[compact operator]]s on the Hilbert space ℓ<sup>2</sup> has a Schauder basis. For every ''x'', ''y'' in ℓ<sup>2</sup>, let {{nowrap|''x'' ⊗ ''y''}} denote the [[Rank (linear algebra)|rank one]] operator {{nowrap| ''v'' &isin; ℓ<sup>2</sup> &rarr; &lt;''v'',&thinsp;''x''&gt;&thinsp;''y''}}. If {{nowrap| {''e''<sub>''n''&thinsp;</sub>}<sub>''n''&thinsp;&ge;&thinsp;1</sub>}} is the standard orthonormal basis of ℓ<sup>2</sup>, a basis for ''K''(ℓ<sup>2</sup>) is given by the sequence<ref name="Ryan">see Proposition&nbsp;4.25, p.&nbsp;88 in {{harvtxt|Ryan|2002}}.</ref>

:<math>\begin{align}  & e_1 \otimes e_1, \ \ e_1 \otimes e_2, \; e_2 \otimes e_2, \; e_2 \otimes e_1, \ldots, \\ 
 & e_1 \otimes e_n, e_2 \otimes e_n, \ldots,  e_n \otimes e_n, e_n \otimes e_{n-1}, \ldots, e_n \otimes e_1, \ldots \end{align}</math>

For every ''n'', the sequence consisting of the ''n''<sup>2</sup> first vectors in this basis is a suitable ordering of the family {''e''<sub>''j''</sub> ⊗ ''e''<sub>''k''</sub>}, for {{nowrap| 1 &le; ''j'', ''k'' &le; ''n''}}.

The preceding result can be generalized: a Banach space ''X'' with a basis has the [[approximation property]], so the space ''K''(''X'') of compact operators on ''X'' is isometrically isomorphic<ref>see Corollary&nbsp;4.13, p.&nbsp;80 in {{harvtxt|Ryan|2002}}.</ref> to the [[Banach space#Tensor product|injective tensor product]]

: <math>X' \widehat \otimes_\varepsilon X \simeq \mathcal{K}(X).</math>

If ''X'' is a Banach space with a Schauder basis {{nowrap| {''e''<sub>''n''&thinsp;</sub>}<sub>''n''&thinsp;&ge;&thinsp;1</sub>}} such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a [[Schauder basis#Schauder bases and duality|shrinking basis]], then the space ''K''(''X'') admits a basis formed by the rank one operators {{nowrap|''e''&thinsp;*<sub>''j''</sub> ⊗ ''e''<sub>''k''</sub> : ''v'' &rarr; ''e''&thinsp;*<sub>''j''&thinsp;</sub>(''v'')&thinsp;''e''<sub>''k''</sub>}}, with the same ordering as before.<ref name="Ryan" />  This applies in particular to every [[Reflexive space|reflexive]] Banach space ''X'' with a Schauder basis

On the other hand, the space ''B''(ℓ<sup>2</sup>) has no basis, since it is non-separable.  Moreover,  ''B''(ℓ<sup>2</sup>) does not have the approximation property.<ref>{{cite journal | last1 = see Szankowski | first1 = Andrzej | year = 1981 | title = ''B''(''H'') does not have the approximation property | url = | journal = Acta Math. | volume = 147 | issue = | pages = 89–108 | doi=10.1007/bf02392870}}</ref>

==Unconditionality==
A Schauder basis {''b''<sub>''n''</sub>} is '''unconditional''' if whenever the series <math> \sum \alpha_nb_n</math> converges, it converges
[[unconditional convergence|unconditionally]].  For a Schauder basis  {''b''<sub>''n''</sub>}, this is equivalent to the existence of a constant ''C'' such that

:<math> \Bigl\| \sum_{k=0}^n \varepsilon_k \alpha_k b_k \Bigr\|_V \le C \Bigl\| \sum_{k=0}^n \alpha_k b_k \Bigr\|_V </math>

for all integers ''n'', all scalar coefficients {&alpha;<sub>''k''</sub>} and all signs {{nowrap| &epsilon;<sub>''k''</sub> {{=}} &plusmn;&thinsp;1}}.
Unconditionality is an important property since it allows one to forget about the order of summation. A Schauder basis is '''symmetric''' if it is unconditional and uniformly equivalent to all its permutations: there exists a constant ''C'' such that for every integer ''n'', every permutation &pi; of the integers {{nowrap| {0, 1, &hellip;, ''n''} }}, all scalar coefficients {&alpha;<sub>''k''</sub>} and all signs&nbsp;{&epsilon;<sub>''k''</sub>},

:<math> \Bigl\| \sum_{k=0}^n \varepsilon_k \alpha_k b_{\pi(k)} \Bigr\|_V \le C \Bigl\| \sum_{k=0}^n \alpha_k b_k \Bigr\|_V. </math>

The standard bases of the [[sequence space]]s ''c''<sub>0</sub> and ℓ<sup>''p''</sup> for 1&nbsp;≤ ''p''&nbsp;< ∞, as well as every orthonormal basis in a Hilbert space, are unconditional. These bases are also symmetric.

The trigonometric system is not an unconditional basis in ''L<sup>p</sup>'', except for ''p''&nbsp;=&nbsp;2.

The Haar system is an unconditional basis in ''L<sup>p</sup>'' for any 1&nbsp;< ''p''&nbsp;< ∞. The space ''L''<sup>1</sup>([0,&nbsp;1]) has no unconditional basis.<ref>see p.&nbsp;24 in {{harvtxt|Lindenstrauss|Tzafriri|1977}}.</ref>

A natural question is whether every infinite-dimensional Banach space has an infinite-dimensional subspace with an unconditional basis. This was solved negatively by [[Timothy Gowers]] and [[Bernard Maurey]] in 1992.<ref>{{Cite arXiv|last = Gowers|first = W. Timothy|author2=Maurey, Bernard|title = The unconditional basic sequence problem |date = 6 May 1992|arxiv = math/9205204}}</ref>

== Schauder bases and duality ==
A basis {''e<sub>n</sub>''}<sub>''n''&ge;0</sub> of a Banach space ''X'' is '''boundedly complete''' if for every sequence {''a<sub>n</sub>''}<sub>''n''&ge;0</sub> of scalars such that the partial sums

:<math> V_n =  \sum_{k=0}^n a_k e_k</math>

are bounded in ''X'', the sequence {''V<sub>n</sub>''} converges in ''X''. The unit vector basis for ℓ<sup>''p''</sup>, {{nowrap|1&nbsp;&le; ''p'' < ∞}}, is boundedly complete. However, the unit vector basis is not boundedly complete in ''c''<sub>0</sub>. Indeed, if ''a<sub>n</sub>''&nbsp;=&nbsp;1 for every ''n'', then

:<math> \|V_n\|_{c_0} = \max_{0 \le k \le n} |a_k| = 1</math>

for every ''n'', but the sequence {''V<sub>n</sub>''} is not convergent in ''c''<sub>0</sub>, since ||''V''<sub>''n''+1</sub> − ''V''<sub>''n''</sub>|| = 1 for every&nbsp;''n''.

A space ''X'' with a boundedly complete basis {''e<sub>n</sub>''}<sub>''n''&ge;0</sub> is [[Banach space|isomorphic]] to a dual space, namely, the space ''X'' is isomorphic to the dual of the closed linear span in the dual {{nowrap|''X''&thinsp;′}} of the biorthogonal functionals associated to the basis {''e<sub>n</sub>''}.<ref>see p.&nbsp;9 in {{harvtxt|Lindenstrauss|Tzafriri|1977}}.</ref>

A basis {''e<sub>n</sub>''}<sub>''n''&ge;0</sub> of ''X'' is '''shrinking''' if for every bounded linear functional ''f'' on ''X'', the sequence of non-negative numbers 
:<math> \varphi_n = \sup \{|f(x)| : x \in F_n, \; \|x\| \le 1 \}</math>
tends to 0 when {{nowrap|''n'' &rarr; ∞}}, where ''F<sub>n</sub>'' is the linear span of the basis vectors ''e<sub>m</sub>'' for ''m'' &ge; ''n''. The unit vector basis for ℓ<sup>''p''</sup>, 1&nbsp;< ''p'' < ∞, or for ''c''<sub>0</sub>, is shrinking. It is not shrinking in ℓ<sup>1</sup>''':''' if ''f'' is the bounded linear functional on ℓ<sup>1</sup> given by 
:<math> f : x = \{x_n\} \in \ell^1 \ \rightarrow \ \sum_{n=0}^{\infty} x_n,</math>
then {{nowrap|&phi;<sub>''n''</sub> &ge; ''f''(''e''<sub>''n''</sub>) {{=}} 1}} for every ''n''.

A basis {{nowrap|&#123;''e''<sub>''n''&thinsp;</sub>&#125;<sub>''n''&thinsp;&ge;&thinsp;0</sub>}} of ''X'' is shrinking if and only if the biorthogonal functionals {{nowrap|&#123;''e''*<sub>''n''&thinsp;</sub>&#125;<sub>''n''&thinsp;&ge;&thinsp;0</sub>}} form a basis of the dual {{nowrap|''X''&thinsp;′}}.<ref>see p.&nbsp;8 in {{harvtxt|Lindenstrauss|Tzafriri|1977}}.</ref>

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.<ref>see James, Robert. C. (1950), "Bases and reflexivity of Banach spaces", ''Ann. of Math.'' (2) '''52''': 518&ndash;527. See also {{harvtxt|Lindenstrauss|Tzafriri|1977}} p.&nbsp;9.</ref>
James also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to ''c''<sub>0</sub> or ℓ<sup>1</sup>.<ref>see James, Robert C. (1950), "Bases and reflexivity of Banach spaces", ''Ann. of Math''. (2) '''52''': 518&ndash;527. See also p.&nbsp;23 in {{harvtxt|Lindenstrauss|Tzafriri|1977}}.</ref>

==Related concepts==
A [[Hamel basis]] is a subset ''B'' of a vector space ''V'' such that every element v ∈ V can uniquely be written as

:<math> v = \sum_{b \in B} \alpha_b b </math>

with α<sub>''b''</sub> ∈ ''F'', with the extra condition that the set

:<math> \{ b \in B \mid \alpha_b \neq 0 \} </math>

is finite.  This property makes the Hamel basis unwieldy for infinite-dimensional Banach spaces; as a Hamel basis for an infinite-dimensional Banach space has to be [[uncountable]]. (Every finite-dimensional subspace of an infinite-dimensional Banach space ''X'' has empty interior, and is no-where dense in ''X''. It then follows from the [[Baire category theorem]] that a countable union of these finite-dimensional subspaces cannot serve as a basis.<ref>Carothers, N. L. (2005), ''A short course on Banach space theory'', Cambridge University Press {{isbn|0-521-60372-2}}</ref>)

==See also==
*[[Generalized Fourier series]]
*[[Orthogonal polynomials]]
*[[Haar wavelet]]
*[[Banach space]]

==Notes==
<references/>
{{PlanetMath attribution|id=1434|title=Countable basis}}

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{{Use dmy dates|date=September 2010}}.
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==Further reading==
*{{Citation
| last= Kufner | first=Alois
| title = Function spaces
| series = De Gruyter Series in Nonlinear analysis and applications
| volume = 14
| publisher =  	Academia Publishing House of the Czechoslovak Academy of Sciences, de Gruyter
| location = 	Prague
| year = 2013
}}

{{Functional Analysis}}

{{DEFAULTSORT:Schauder Basis}}
[[Category:Banach spaces]]