Series summability of complete biorthogonal sequences

Abstract

A complete biorthogonal sequence in a topological vector space X is a double sequence (x_i, f_i) such that (1) each x_i ∈ X and each f_i ∈ X* the dual space of X ; (2) the closed linear span of {x_i} is dense in X, (3) f_i (x_j) = 1 if i = j and is 0 otherwise, and (4) f_i (z) = 0 for each i only when z = 0. If x ∈ X and f ∈ X* the numerical series

$${{\sum\limits_{i} f }_{i}}(x)f({{x}_{i}}) $$

will converge to f(x) if (a) x is a finite linear combination of (x_i); (b) f is a.finite linear combination of (f_i) or (c) (x_i) is a Schauder basis of X. In other cases (1) may not converge, or z if it does it may not converge to f (x); but there may be some method of series summability that will associate the “correct” sum, i. e. f (x), to the sequence (f_i(x) f (x_i)). In order to determine conditions under which there is such a method of summability we study four sequence spaces associated with the biorthogonal sequence (x_i, f_i), namely S = {(f_i (x)): x ∈ X}, that represents the space X,S^f= { (f(x_i)) : f ∈ X*}, that represents the dual space X* of X, S(S), the series space of S that consists of the linear span of all sequences of the form st where s ∈ S and t ∈ S^f, the multiplier space M (S) consisting of all sequences u such that US ∈ S whenever s ∈ S. We will discuss how conditions on these four spaces result in summability properties of the biorthogonal sequence (x_i, f_i) and also on the topology of the space X.