Schauder basis

Let V denote a Banach space over the field F. A Schauder basis is a sequence {b_n} of elements of V such that for every element v ∈ V there exists a unique sequence {α_n} of scalars in F so that

${\displaystyle v=\sum _{n=0}^{\infty }\alpha _{n}b_{n},}$

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

${\displaystyle \lim _{n\to \infty }\left\|v-\sum _{k=0}^{n}\alpha _{k}b_{k}\right\|_{V}=0.}$

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 unconditionally.

A Schauder basis {b_n}_n ≥ 0 is said to be normalized when all the basis vectors have norm 1 in the Banach space V.

A sequence {x_n}_n ≥ 0 in V is a basic sequence if it is a Schauder basis of its 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 c > 0 and C such that for every natural number N ≥ 0 and all sequences {α_n} of scalars,

${\displaystyle c\left\|\sum _{k=0}^{N}\alpha _{k}b_{k}\right\|_{V}\leq \left\|\sum _{k=0}^{N}\alpha _{k}c_{k}\right\|_{W}\leq C\left\|\sum _{k=0}^{N}\alpha _{k}b_{k}\right\|_{V}.}$

A family of vectors in V is total if its linear span (the set of finite linear combinations) is 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 1, then B is an orthonormal basis of V.

Let {b_n} 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 {P_n} defined by

${\displaystyle v=\sum _{k=0}^{\infty }\alpha _{k}b_{k}\ \ {\overset {\textstyle P_{n}}{\longrightarrow }}\ \ P_{n}(v)=\sum _{k=0}^{n}\alpha _{k}b_{k}}$

are uniformly bounded by some constant C.[4] When C = 1, the basis is called a monotone basis. The maps {P_n} are the 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,

${\displaystyle |b_{n}^{*}(v)|\;\|b_{n}\|_{V}=|\alpha _{n}|\;\|b_{n}\|_{V}=\|\alpha _{n}b_{n}\|_{V}=\|P_{n}(v)-P_{n-1}(v)\|_{V}\leq 2C\|v\|_{V}.}$

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 ≤ 2C in the continuous dual V ′ 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[5] 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.[6]

The standard unit vector bases of c₀, and of ℓ^p for 1 ≤ p < ∞, are monotone Schauder bases. In this unit vector basis {b_n}, the vector b_n in V = c₀ or in V = ℓ^p is the scalar sequence {b_n, j}_j where all coordinates b_{n, j} are 0, except the nth coordinate:

${\displaystyle b_{n}=\{b_{n,j}\}_{j=0}^{\infty }\in V,\ \ b_{n,j}=\delta _{n,j},}$

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.

The Haar system is an example of a basis for L^p([0, 1]), when 1 ≤ p < ∞.[2] When 1 < p < ∞, another example is the trigonometric system defined below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis. The Faber–Schauder system is the most commonly used Schauder basis for C([0, 1]).[3][7]

Several bases for classical spaces were discovered before Banach's book appeared (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 Franklin system exists in A(D).[8] One can also prove that the periodic Franklin system[9] is a basis for a Banach space A_r isomorphic to A(D).[10] 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([0, 1]),[11] and it is a Schauder basis in L^p([0, 1]) when 1 ≤ p < ∞.[12] Systems derived from the Franklin system give bases in the space C¹([0, 1]²) of differentiable functions on the unit square.[13] The existence of a Schauder basis in C¹([0, 1]²) was a question from Banach's book.[14]

A Schauder basis {b_n} is unconditional if whenever the series ${\displaystyle \sum \alpha _{n}b_{n}}$ converges, it converges unconditionally. For a Schauder basis {b_n}, this is equivalent to the existence of a constant C such that

${\displaystyle {\Bigl \|}\sum _{k=0}^{n}\varepsilon _{k}\alpha _{k}b_{k}{\Bigr \|}_{V}\leq C{\Bigl \|}\sum _{k=0}^{n}\alpha _{k}b_{k}{\Bigr \|}_{V}}$

for all natural numbers n, all scalar coefficients {α_k} and all signs 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 permutations: there exists a constant C such that for every natural number n, every permutation π of the set {0, 1, …, n}, all scalar coefficients {α_k} and all signs {ε_k},

${\displaystyle {\Bigl \|}\sum _{k=0}^{n}\varepsilon _{k}\alpha _{k}b_{\pi (k)}{\Bigr \|}_{V}\leq C{\Bigl \|}\sum _{k=0}^{n}\alpha _{k}b_{k}{\Bigr \|}_{V}.}$

The standard bases of the sequence spaces c₀ 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 L¹([0, 1]) has no unconditional basis.[19]

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.[20]

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

${\displaystyle V_{n}=\sum _{k=0}^{n}a_{k}e_{k}}$

are bounded in X, the sequence {V_n} converges in X. The unit vector basis for ℓ^p, 1 ≤ p < ∞, is boundedly complete. However, the unit vector basis is not boundedly complete in c₀. Indeed, if a_n = 1 for every n, then

${\displaystyle \|V_{n}\|_{c_{0}}=\max _{0\leq k\leq n}|a_{k}|=1}$

for every n, but the sequence {V_n} is not convergent in c₀, since ||V_n+1 − V_n|| = 1 for every n.

A space X with a boundedly complete basis {e_n}_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 X ′ of the biorthogonal functionals associated to the basis {e_n}.[21]

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

${\displaystyle \varphi _{n}=\sup\{|f(x)|:x\in F_{n},\;\|x\|\leq 1\}}$

tends to 0 when 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 < p < ∞, or for c₀, is shrinking. It is not shrinking in ℓ¹: if f is the bounded linear functional on ℓ¹ given by

${\displaystyle f:x=\{x_{n}\}\in \ell ^{1}\ \rightarrow \ \sum _{n=0}^{\infty }x_{n},}$

then φ_n ≥ f(e_n) = 1 for every n.

A basis {e_n}_n ≥ 0 of X is shrinking if and only if the biorthogonal functionals {e*_n}_n ≥ 0 form a basis of the dual X ′.[22]

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.[23] James also proved that a space with an unconditional basis is non-reflexive if and only if it contains a subspace isomorphic to c₀ or ℓ¹.[24]

A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as

${\displaystyle v=\sum _{b\in B}\alpha _{b}b}$

with α_b ∈ F, with the extra condition that the set

${\displaystyle \{b\in B\mid \alpha _{b}\neq 0\}}$

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.[25])

• Markushevich basis
• Generalized Fourier series
• Orthogonal polynomials
• Haar wavelet
• Banach space

This article incorporates material from Countable basis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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