In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; 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 functions on an interval, sometimes called a Faber–Schauder system.
Definitions 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 v = ∑ n = 0 ∞ α n b n . {\displaystyle v=\sum _{n=0}^{\infty }{\alpha _{n}b_{n}}{\text{.}}} The convergence of the infinite sum is implicitly that of the ambient topology, i.e., lim n → ∞ ∑ k = 0 n α k b k = v , {\displaystyle \lim _{n\to \infty }{\sum _{k=0}^{n}\alpha _{k}b_{k}}=v{\text{,}}} 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. 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 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 {bn}n ≥ 0 is said to be normalized when all the basis vectors have norm 1 in the Banach space V. A sequence {xn}n ≥ 0 in V is a basic sequence if it is a Schauder basis of its closed linear span. 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,
c ‖ ∑ k = 0 N α k b k ‖ V ≤ ‖ ∑ k = 0 N α k c k ‖ W ≤ C ‖ ∑ k = 0 N α k b k ‖ V . {\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.
Properties Let {bn} 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 {Pn} defined by
… excerpt ends here. Continue reading the full article.
