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Schauder bases

A Schauder basis in a Banach space X {\displaystyle X} is a sequence { e n } n ≥ 0 {\displaystyle \left\{e_{n}\right\}_{n\geq 0}} of vectors in X {\displaystyle X} with the property that for every vector x ∈ X, {\displaystyle x\in X,} there exist uniquely defined scalars { x n } n ≥ 0 {\displaystyle \left\{x_{n}\right\}_{n\geq 0}} depending on x, {\displaystyle x,} such that x = ∑ n = 0 ∞ x n e n, i.e., x = lim n P n (x), P n (x):= ∑ k = 0 n x k e k. {\displaystyle x=\sum _{n=0}^{\infty }x_{n}e_{n},\quad {\textit {i.e.,}}\quad x=\lim _{n}P_{n}(x),\ P_{n}(x):=\sum _{k=0}^{n}x_{k}e_{k}.}

Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.

It follows from the Banach–Steinhaus theorem that the linear mappings { P n } {\displaystyle \left\{P_{n}\right\}} are uniformly bounded by some constant C. {\displaystyle C.} Let { e n ∗ } {\displaystyle \left\{e_{n}^{*}\right\}} denote the coordinate functionals which assign to every x {\displaystyle x} in X {\displaystyle X} the coordinate x n {\displaystyle x_{n}} of x {\displaystyle x} in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm 1, {\displaystyle 1,} the coordinate functionals { e n ∗ } {\displaystyle \left\{e_{n}^{*}\right\}} have norm ≤ 2 C {\displaystyle \,\leq 2C} in the dual of X. {\displaystyle X.}

Most classical separable spaces have explicit bases. The Haar system { h n } {\displaystyle \left\{h_{n}\right\}} is a basis for L p (), 1 ≤ p < ∞. {\displaystyle L^{p}(),1\leq p<\infty.} The trigonometric system is a basis in L p (T) {\displaystyle L^{p}(\mathbf {T})} when 1 < p < ∞. {\displaystyle 1<p<\infty.} The Schauder system is a basis in the space C (). {\displaystyle C().} The question of whether the disk algebra A (D) {\displaystyle A(\mathbf {D})} has a basis remained open for more than forty years, until Bočkarev showed in 1974 that A (D) {\displaystyle A(\mathbf {D})} admits a basis constructed from the Franklin system.

Since every vector x {\displaystyle x} in a Banach space X {\displaystyle X} with a basis is the limit of P n (x), {\displaystyle P_{n}(x),} with P n {\displaystyle P_{n}} of finite rank and uniformly bounded, the space X {\displaystyle X} satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.

Robert C. James characterized reflexivity in Banach spaces with a basis: the space X {\displaystyle X} with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. In this case, the biorthogonal functionals form a basis of the dual of X. {\displaystyle X.}