Schauder basis

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. 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 G. Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called a Faber–Schauder system. Let V denote a Banach 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 where the convergence is understood with respect to the norm topology, i.e., 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 {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 integer 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. Further, when each element in B has norm 1, then B is an orthonormal basis of V.