In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Schauder basis or countable basis is similar to the usual (
Hamel)
basis of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
; the difference is that Hamel bases use
linear combinations 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 spaces including
Banach spaces.
Schauder bases were described by
Juliusz Schauder in 1927,
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
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s on an
interval, sometimes called a Faber–Schauder system.
[Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) 19: 104–112. ;
http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553]
Definitions
Let ''V'' denote a
topological vector space over the
field ''F''. A Schauder basis is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of elements of ''V'' such that for every element there exists a ''unique'' sequence of scalars in ''F'' so that
The convergence of the infinite sum is implicitly that of the ambient topology, ''i.e.'',
but can be reduced to only
weak convergence in a
normed vector space (such as a
Banach space). Unlike a
Hamel basis, the elements of the basis must be ordered since the series may not converge
unconditionally
"Unconditionally" is a song by American singer Katy Perry. It was released as the second single from her fourth studio album ''Prism'' (2013) on October 16, 2013, two days before the album was released. Inspiration for the song came primarily fro ...
.
Though the definition above technically does not require a normed space, a norm is necessary to do say almost anything useful about Schauder bases. The results below assume the existence of a norm.
A Schauder basis is said to be normalized when all the basis vectors have norm 1 in the Banach space ''V''.
A sequence in ''V'' is a basic sequence if it is a Schauder basis of its
closed linear span.
Two Schauder bases, in ''V'' and in ''W'', are said to be equivalent if there exist two constants and ''C'' such that for every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
and all sequences of scalars,
:
A family of vectors in ''V'' is total if its
linear span (the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of finite linear combinations) is
dense in ''V''. If ''V'' is a
Hilbert space, an
orthogonal basis is a ''total''
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''B'' of ''V'' such that elements in ''B'' are nonzero and pairwise orthogonal. Further, when each element in ''B'' has norm 1, then ''B'' is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of ''V''.
Properties
Let be a Schauder basis of a Banach space ''V'' over F = R or C. It is a subtle consequence of the
open mapping theorem that the linear mappings defined by
:
are uniformly bounded by some constant ''C''. When , the basis is called a monotone basis. The maps are the
basis projections.
Let 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'',
:
These functionals are called biorthogonal functionals associated to the basis . When the basis is normalized, the coordinate functionals have norm ≤ 2''C'' in the
continuous dual of ''V''.
A Banach space with a Schauder basis is necessarily
separable, but the converse is false. 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
bounded approximation property.
A theorem attributed to
Mazur 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.
Examples
The standard unit vector bases of
''c''0, and of
ℓ''p'' for 1 ≤ ''p'' < ∞, are monotone Schauder bases. In this unit vector basis , the vector ''b
n'' in or in is the scalar sequence where all coordinates ''b
n, j'' are 0, except the ''n''th coordinate:
:
where δ
''n, j'' is the
Kronecker delta. The space ℓ
∞ is not separable, and therefore has no Schauder basis.
Every
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
in a separable
Hilbert space is a Schauder basis. Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ
2.
The
Haar system is an example of a basis for
''L''''p''( ">, 1, when 1 ≤ ''p'' < ∞.
When , another example is the trigonometric system defined below. The Banach space ''C''(
, 1 of continuous functions on the interval
, 1 with the
supremum norm, admits a Schauder basis. The
Faber–Schauder system is the most commonly used Schauder basis for ''C''(
, 1.
Several bases for classical spaces were discovered before Banach's book appeared (), 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
Franklin system exists in ''A''(''D''). One can also prove that the periodic Franklin system is a basis for a Banach space ''A''
''r'' isomorphic to ''A''(''D'').
This space ''A''
''r'' consists of all complex continuous functions on the unit circle T whose
conjugate function is also continuous. The Franklin system is another Schauder basis for ''C''(
, 1,
and it is a Schauder basis in ''L''
''p''(
, 1 when .
Systems derived from the Franklin system give bases in the space ''C''
1(
, 1sup>2) of
differentiable functions on the unit square. The existence of a Schauder basis in ''C''
1(
, 1sup>2) was a question from Banach's book.
Relation to Fourier series
Let be, in the real case, the sequence of functions
:
or, in the complex case,
:
The sequence is called the trigonometric system. It is a Schauder basis for the space
''L''''p''( ">, 2''π'' for any ''p'' such that . For ''p'' = 2, this is the content of the
Riesz–Fischer theorem, and for ''p'' ≠ 2, it is a consequence of the boundedness on the space ''L''
''p''(
, 2''π'' of the
Hilbert transform on the circle. It follows from this boundedness that the projections ''P''
''N'' defined by
:
are uniformly bounded on ''L''
''p''(
, 2''π'' when . This family of maps is
equicontinuous and tends to the identity on the dense subset consisting of
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
s. It follows that ''P''
''N''''f'' tends to ''f'' in ''L''
''p''-norm for every . In other words, is a Schauder basis of ''L''
''p''(
, 2''π''.
However, the set is not a Schauder basis for ''L''
1(
, 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 is not a Schauder basis for ''C''(
, 2''π''.
Bases for spaces of operators
The space ''K''(ℓ
2) of
compact operators on the Hilbert space ℓ
2 has a Schauder basis. For every ''x'', ''y'' in ℓ
2, let denote the
rank one operator . If is the standard orthonormal basis of ℓ
2, a basis for ''K''(ℓ
2) is given by the sequence
[see Proposition 4.25, p. 88 in .]
:
For every ''n'', the sequence consisting of the ''n''
2 first vectors in this basis is a suitable ordering of the family , for .
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 to the
injective tensor product In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so it ...
:
If ''X'' is a Banach space with a Schauder basis such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a
shrinking basis, then the space ''K''(''X'') admits a basis formed by the rank one operators , with the same ordering as before.
This applies in particular to every
reflexive Banach space ''X'' with a Schauder basis
On the other hand, the space ''B''(ℓ
2) has no basis, since it is non-separable. Moreover, ''B''(ℓ
2) does not have the approximation property.
Unconditionality
A Schauder basis is unconditional if whenever the series
converges, it converges
unconditionally
"Unconditionally" is a song by American singer Katy Perry. It was released as the second single from her fourth studio album ''Prism'' (2013) on October 16, 2013, two days before the album was released. Inspiration for the song came primarily fro ...
. For a Schauder basis , this is equivalent to the existence of a constant ''C'' such that
:
for all natural numbers ''n'', all scalar coefficients and all signs .
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 natural number ''n'', every permutation π of the set , all scalar coefficients and all signs ,
:
The standard bases of the
sequence spaces ''c''
0 and ℓ
''p'' for 1 ≤ ''p'' < ∞, 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'' = 2.
The Haar system is an unconditional basis in ''L
p'' for any 1 < ''p'' < ∞. The space ''L''
1(
, 1 has no unconditional basis.
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.
Schauder bases and duality
A basis
''n''≥0 of a Banach space ''X'' is boundedly complete if for every sequence
''n''≥0 of scalars such that the partial sums
:
are bounded in ''X'', the sequence converges in ''X''. The unit vector basis for ℓ
''p'', , is boundedly complete. However, the unit vector basis is not boundedly complete in ''c''
0. Indeed, if ''a
n'' = 1 for every ''n'', then
:
for every ''n'', but the sequence is not convergent in ''c''
0, since , , ''V''
''n''+1 − ''V''
''n'', , = 1 for every ''n''.
A space ''X'' with a boundedly complete basis
''n''≥0 is
isomorphic to a dual space, namely, the space ''X'' is isomorphic to the dual of the closed linear span in the dual of the biorthogonal functionals associated to the basis .
A basis
''n''≥0 of ''X'' is shrinking if for every bounded linear functional ''f'' on ''X'', the sequence of non-negative numbers
:
tends to 0 when , where ''F
n'' is the linear span of the basis vectors ''e
m'' for ''m'' ≥ ''n''. The unit vector basis for ℓ
''p'', 1 < ''p'' < ∞, or for ''c''
0, is shrinking. It is not shrinking in ℓ
1: if ''f'' is the bounded linear functional on ℓ
1 given by
:
then for every ''n''.
A basis of ''X'' is shrinking if and only if the biorthogonal functionals form a basis of the dual .
Robert C. James Robert Clarke James (1918 – September 25, 2004) was an Americans, American mathematician who worked in functional analysis.
Biography
James attended UCLA as an undergraduate, where his father was a professor. As a devout Quakers, Quaker, he was ...
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.
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 ℓ
1.
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
:
with α
''b'' ∈ ''F'', with the extra condition that the set
:
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
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
that a countable union of these finite-dimensional subspaces cannot serve as a basis.
[Carothers, N. L. (2005), ''A short course on Banach space theory'', Cambridge University Press ])
See also
*
Markushevich basis In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both ''complete'' and ''total''.
Definition
Let X be Banach space. A biorthogonal system In mathematics, a biorthogonal system is a pair of in ...
*
Generalized Fourier series
*
Orthogonal polynomials
*
Haar wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repre ...
*
Banach space
Notes
References
* .
*
*
*
*
*
* .
* .
*
.
* .
*Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655
Further reading
*
{{DEFAULTSORT:Schauder Basis
Banach spaces