Schauder basis

{{Short description|Computational tool}} 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 | journal = [[Mathematische Zeitschrift]] | volume = 28 | 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]] discussed in 1910 a basis for [[continuous function]]s on an [[Interval (mathematics)|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 [[topological vector space]] over the [[field (mathematics)|field]]&nbsp;''F''. A '''Schauder basis''' is a [[sequence]] {''b''_''n''} of elements of&nbsp;''V'' such that for every element {{nowrap|''v'' ∈ ''V''}} there exists a ''unique'' sequence {α_''n''} of scalars in&nbsp;''F'' so that <math display=block>v = \sum_{n=0}^\infty{\alpha_nb_n}\text{.}</math> The convergence of the infinite sum is implicitly that of the ambient topology, ''i.e.'', <math display=block> \lim_{n \to \infty}{\sum_{k=0}^n \alpha_k b_k}=v\text{,}</math> but can be reduced to only [[weak topology|weak convergence]] in a [[normed vector space]] (such as a [[Banach space]]).<ref>{{Cite journal |last=Karlin |first=S. |date=December 1948 |title=Bases in Banach spaces |url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-15/issue-4/Bases-in-Banach-spaces/10.1215/S0012-7094-48-01587-7.full |journal=[[Duke Mathematical Journal]] |volume=15 |issue=4 |pages=971–985 |doi=10.1215/S0012-7094-48-01587-7 |issn=0012-7094|url-access=subscription }}</ref> Unlike a [[Hamel basis]], the elements of the basis must be ordered, since the series may not converge [[unconditional convergence|unconditionally]].

Note that some authors define Schauder bases to be countable (as above), while others use the term to include uncountable bases. In either case, the sums themselves always are countable. An uncountable Schauder basis is a [[linearly ordered set]] rather than a sequence, and each sum inherits the order of its terms from this linear ordering. They can and do arise in practice. As an example, a [[separable space|separable]] [[Hilbert space]] can only have a countable Schauder basis, but a non-separable Hilbert space may have an uncountable one.

Though the definition above technically does not require a normed space, a norm is necessary to say almost anything useful about Schauder bases. The results below assume the existence of a norm.

A Schauder basis {{nowrap|{''b''_''n''}_{''n'' ≥ 0}}} is said to be '''normalized''' when all the basis vectors have norm&nbsp;1 in the Banach space&nbsp;''V''.

A sequence {{nowrap|{''x''_''n''}_{''n'' ≥ 0}}} in ''V'' is a '''basic sequence''' if it is a Schauder basis of its [[Linear span#Closed linear span (functional analysis)|closed linear span]].

Two Schauder bases, {''b''_''n''} in ''V'' and {''c''_''n''} in ''W'', are said to be '''equivalent''' if there exist two constants {{nowrap|''c'' > 0}} and ''C'' such that for every [[natural number]] {{nowrap|''N'' ≥ 0}} and all sequences {α_''n''} 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 (mathematics)|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_n''} 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_n''} 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>{{harvtxt|Fabian|Habala|Hájek|Montesinos|2011|loc=Theorem 4.10}}</ref> When {{nowrap|''C'' {{=}} 1}}, the basis is called a '''monotone''' basis. The maps {''P_n''} are the '''[[Projection (linear algebra)|basis projections]]'''.

Let {''b*_n''} denote the '''coordinate functionals''', where ''b*_n'' assigns to every vector ''v'' in ''V'' the coordinate α_''n'' of ''v'' in the above expansion. Each ''b*_n'' 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*_n''} are called '''biorthogonal functionals''' associated to the basis {''b''_''n''}. When the basis {''b''_''n''} is normalized, the coordinate functionals {''b*_n''} have norm ≤ 2''C'' in the [[Dual space#Continuous dual space|continuous dual]] {{nowrap|''V''{{hair space}}′}} of&nbsp;''V''.

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

A Banach space with a Schauder basis is necessarily [[Separable space|separable]], but the converse is false. 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 [[Reflexive Banach space|reflexive]] and separable Banach space failing the approximation property, thus a space without a Schauder basis.<ref>{{Cite journal| last=Enflo | first=Per | author-link=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 | doi-access=free }}</ref> The construction has been simplified and generalized over the years. See {{harvtxt|Fabian|Habala|Hájek|Montesinos|2011|loc=Sec. 16.5}} for a modern presentation.

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.

== Examples == The standard [[unit vector]] bases of [[sequence space|''c''₀]], and of [[sequence space|ℓ^''p'']] for 1&nbsp;≤ ''p''&nbsp;< ∞, are monotone Schauder bases. In this '''unit vector basis''' {''b_n''}, the vector ''b_n'' in {{nowrap|''V'' {{=}} ''c''₀}} or in {{nowrap|''V'' {{=}} ℓ^''p''}} is the scalar sequence {{nowrap|[''b''_{''n'', ''j''}]_''j''}} where all coordinates ''b_{n, j}'' 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 δ_{''n, j''} is the [[Kronecker delta]]. The space ℓ^∞ 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 ℓ².

Every [[Auerbach basis]] in a separable Banach space is a Schauder basis.

The [[Haar wavelet|Haar system]] is an example of a basis for [[Lp space|''L''^''p''([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 monotone 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, §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''_''r'' isomorphic to ''A''(''D'').<ref>See p.&nbsp;192, III.E.17 in {{harvtxt|Wojtaszczyk|1991}}.</ref>

This space ''A''_''r'' 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 | journal = [[Math. Ann.]] | volume = 100 | pages = 522–529 | doi=10.1007/bf01448860}}</ref> and it is a Schauder basis in ''L''^''p''([0,&nbsp;1]) when {{nowrap|1 ≤ ''p'' &lt; ∞}}.<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''¹([0,&nbsp;1]²) of [[Differentiable function|differentiable]] functions on the [[unit square]].<ref>{{cite journal | last1 = see Ciesielski | first1 = Z | year = 1969 | title = A construction of basis in ''C''¹(''I''²) | journal = [[Studia Math.]] | volume = 33 | pages = 243–247 }} and {{cite journal | last1 = Schonefeld | first1 = Steven | year = 1969 | title = Schauder bases in spaces of differentiable functions | journal = [[Bull. Amer. Math. Soc.]] | volume = 75 | issue = 3| pages = 586–590 | doi=10.1090/s0002-9904-1969-12249-4| doi-access = free }}</ref> The existence of a Schauder basis in ''C''¹([0,&nbsp;1]²) was a question from Banach's book.<ref>see p.&nbsp;238, §3 in {{harvtxt|Banach|1932}}.</ref>

=== Relation to Fourier series === Let {''x''_''n''} 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''_''n''} is called the '''trigonometric system'''. It is a Schauder basis for the space [[Lp space|''L''^''p''([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;≠&nbsp;2, it is a consequence of the boundedness on the space ''L''^''p''([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''_''N'' 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''^''p''([0,&nbsp;2''π'']) when {{nowrap|1 &lt; ''p'' &lt; ∞}}. This family of maps {''P''_''N''} is [[Equicontinuity|equicontinuous]] and tends to the identity on the dense subset consisting of [[trigonometric polynomial]]s. It follows that ''P''_''N''''f'' tends to ''f'' in ''L''^''p''-norm for every {{nowrap| ''f'' ∈ ''L''^''p''([0,&nbsp;2''π''])}}. In other words, {''x''_''n''} is a Schauder basis of ''L''^''p''([0,&nbsp;2''π'']).<ref>see p.&nbsp;40, II.B.11 in {{harvtxt|Wojtaszczyk|1991}}.</ref>

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

=== Bases for spaces of operators === The space ''K''(ℓ²) of [[compact operator]]s on the Hilbert space ℓ² has a Schauder basis. For every ''x'', ''y'' in ℓ², let {{nowrap|''x'' ⊗ ''y''}} denote the [[Rank (linear algebra)|rank one]] operator {{nowrap| ''v'' ∈ ℓ² → &lt;''v'', ''x'' &gt; ''y''}}. If {{nowrap| {''e''_''n''}_{''n'' ≥ 1}}} is the standard orthonormal basis of ℓ², a basis for ''K''(ℓ²) 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''² first vectors in this basis is a suitable ordering of the family {''e''_''j'' ⊗ ''e''_''k''}, for {{nowrap| 1 ≤ ''j'', ''k'' ≤ ''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''_''n''}_{''n'' ≥ 1}}} 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''*_''j'' ⊗ ''e''_''k'' : ''v'' → ''e''*_''j''(''v'') ''e''_''k''}}, 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''(ℓ²) has no basis, since it is non-separable. Moreover, ''B''(ℓ²) 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 | journal = Acta Math. | volume = 147 | pages = 89–108 | doi=10.1007/bf02392870| doi-access = free }}</ref>

==Unconditionality== A Schauder basis {''b''_''n''} is '''unconditional''' if whenever the series <math> \sum \alpha_nb_n</math> converges, it converges [[unconditional convergence|unconditionally]]. For a Schauder basis {''b''_''n''}, 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 natural numbers ''n'', all scalar coefficients {α_''k''} and all signs {{nowrap| ε_''k'' {{=}} ±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 [[permutation]]s: there exists a constant ''C'' such that for every natural number ''n'', every permutation π of the set {{nowrap| {0, 1, ..., ''n''}}}, all scalar coefficients {α_''k''} and all signs&nbsp;{ε_''k''},

:<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''₀ and ℓ^''p'' 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^p'', except for ''p''&nbsp;=&nbsp;2.

The Haar system is an unconditional basis in ''L^p'' for any 1&nbsp;< ''p''&nbsp;< ∞. The space ''L''¹([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|eprint = math/9205204 }}</ref>

== Schauder bases and duality == A basis {''e_n''}_''n''≥0 of a Banach space ''X'' is '''boundedly complete''' if for every sequence {''a_n''}_''n''≥0 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_n''} converges in ''X''. The unit vector basis for ℓ^''p'', {{nowrap|1 ≤ ''p'' < ∞}}, is boundedly complete. However, the unit vector basis is not boundedly complete in ''c''₀. Indeed, if ''a_n''&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_n''} is not convergent in ''c''₀, since ||''V''_''n''+1 − ''V''_''n''|| = 1 for every&nbsp;''n''.

A space ''X'' with a boundedly complete basis {''e_n''}_''n''≥0 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''{{hair space}}′}} of the biorthogonal functionals associated to the basis {''e_n''}.<ref>see p.&nbsp;9 in {{harvtxt|Lindenstrauss|Tzafriri|1977}}.</ref>

A basis {''e_n''}_''n''≥0 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'' → ∞}}, where ''F_n'' is the linear span of the basis vectors ''e_m'' for ''m'' ≥ ''n''. The unit vector basis for ℓ^''p'', 1&nbsp;< ''p'' < ∞, or for ''c''₀, is shrinking. It is not shrinking in ℓ¹''':''' if ''f'' is the bounded linear functional on ℓ¹ given by :<math> f : x = \{x_n\} \in \ell^1 \ \rightarrow \ \sum_{n=0}^{\infty} x_n,</math> then {{nowrap|φ_''n'' ≥ ''f''(''e''_''n'') {{=}} 1}} for every ''n''.

A basis {{nowrap|[''e''_''n'']_{''n'' ≥ 0}}} of ''X'' is shrinking if and only if the biorthogonal functionals {{nowrap|[''e''*_''n'']_{''n'' ≥ 0}}} form a basis of the dual {{nowrap|''X''{{hair space}}′}}.<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 {{harvtxt|James|1950}} and {{harvtxt|Lindenstrauss|Tzafriri|1977|p=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''₀ or ℓ¹.<ref>See {{harvtxt|James|1950}} and {{harvtxt|Lindenstrauss|Tzafriri|1977|p=23}}.</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''_''b'' ∈ ''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 (topology)|interior]], and is [[nowhere dense]] in ''X''. It then follows from the [[Baire category theorem]] that a countable union of bases 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== *[[Markushevich basis]] *[[Generalized Fourier series]] *[[Orthogonal polynomials]] *[[Haar wavelet]] *[[Banach space]]

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

== References == * {{citation | last = Schauder |first = Juliusz | author-link = Juliusz Schauder | title = Zur Theorie stetiger Abbildungen in Funktionalraumen | language = de | journal = Mathematische Zeitschrift | volume = 26 | year = 1927 | pages = 47&ndash;65 | doi=10.1007/BF01475440|hdl = 10338.dmlcz/104881 | hdl-access = free }}. * {{Banach Théorie des Opérations Linéaires}} <!-- {{sfn | Banach | 1932 | p=}} --> * {{citation | last1 = Fabian | first1 = Marián | last2 = Habala | first2 = Petr | last3 = Hájek | first3 = Petr | last4 = Montesinos | first4 = Vicente | last5 = Zizler | first5 = Václav | authorlink5 = Václav Zizler | title = Banach Space Theory: The Basis for Linear and Nonlinear Analysis | year = 2011 | publisher = Springer | series = CMS Books in Mathematics | isbn = 978-1-4419-7514-0}} * {{cite journal |last1=James |first1=Robert C. |author-link1=Robert C. James |title=Bases and reflexivity of Banach spaces |journal=The Annals of Mathematics |year=1950 |volume=52 |issue=3 |pages=518–527 |doi=10.2307/1969430 |jstor=1969430 }} {{MathSciNet|id=39915}} * {{citation | last1=Lindenstrauss | first1=Joram |author1-link = Joram Lindenstrauss | last2=Tzafriri | first2=Lior | isbn = 3-540-08072-4 | location = Berlin | publisher = Springer-Verlag | series = Ergebnisse der Mathematik und ihrer Grenzgebiete | title=Classical Banach Spaces I, Sequence Spaces | volume = 92 | year=1977}} * {{citation | last = Ryan |first = Raymond A. | year = 2002 | title = Introduction to Tensor Products of Banach Spaces | publisher = Springer-Verlag | series = Springer Monographs in Mathematics | location = London | isbn = 1-85233-437-1 | pages = xiv+225 }} * {{Citation | last = Schaefer | first = Helmut H. | year = 1971 | title = Topological vector spaces | series = [[Graduate Texts in Mathematics]] | volume = 3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6 | pages = xi+294}}. * {{citation | last=Wojtaszczyk | first= Przemysław | title = Banach spaces for analysts | series = Cambridge Studies in Advanced Mathematics | volume = 25 | publisher = Cambridge University Press | location = Cambridge | year= 1991 | pages = xiv+382 | isbn = 0-521-35618-0 }}. * {{eom|id=f/f038020 |title=Faber–Schauder system|first=B.I.|last= Golubov}} {{Use dmy dates|date=April 2020}}. * {{Cite web | last=Heil | first=Christopher E. | title=A basis theory primer | url=http://www.math.gatech.edu/~heil/papers/bases.pdf | year=1997 }}. *Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655

==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 }}

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