[3] Let V denote a topological vector space over the field F. A Schauder basis is a sequence {bn} of elements of V such that for every element v ∈ V there exists a unique sequence {αn} of scalars in F so that
Two Schauder bases, {bn} in V and {cn} 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 {αn} of scalars, 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.
When the basis {bn} is normalized, the coordinate functionals {b*n} have norm ≤ 2C in the continuous dual V ′ of V. Since every vector v in a Banach space V with a Schauder basis is the limit of Pn(v), with Pn of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property.
A Banach space with a Schauder basis is necessarily separable, but the converse is false.
The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis.
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).
[9] One can also prove that the periodic Franklin system[10] is a basis for a Banach space Ar isomorphic to A(D).
[14] The existence of a Schauder basis in C1([0, 1]2) was a question from Banach's book.
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 Lp([0, 2π]) of the Hilbert transform on the circle.
It follows from this boundedness that the projections PN defined by are uniformly bounded on Lp([0, 2π]) when 1 < p < ∞.
This family of maps {PN} is equicontinuous and tends to the identity on the dense subset consisting of trigonometric polynomials.
This means that there are functions in L1 whose Fourier series does not converge in the L1 norm, or equivalently, that the projections PN are not uniformly bounded in L1-norm.
[17] This applies in particular to every reflexive Banach space X with a Schauder basis.
For a Schauder basis {bn}, this is equivalent to the existence of a constant C such that for all natural numbers n, all scalar coefficients {αk} and all signs εk = ±1.
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 {αk} and all signs {εk}, The standard bases of the sequence spaces c0 and ℓp for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert space, are unconditional.
Indeed, if an = 1 for every n, then for every n, but the sequence {Vn} is not convergent in c0, since ||Vn+1 − Vn|| = 1 for every n. A space X with a boundedly complete basis {en}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 {en}.
It is not shrinking in ℓ1: if f is the bounded linear functional on ℓ1 given by then φn ≥ f(en) = 1 for every n. A basis [en]n ≥ 0 of X is shrinking if and only if the biorthogonal functionals [e*n]n ≥ 0 form a basis of the dual X ′.
[24] 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.
[25] A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as with αb ∈ 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 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.