Series summability of complete biorthogonal sequences

A complete biorthogonal sequence in a topological vector space X is a double sequence (x(i), f(i)) such that (1) each x(i) is an element of X and each f(i) is an element of 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 is an element of X and f is an element of X* the numerical series Sigma 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 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 is an element of X), that represents the space X, S-f = {(f (x(i))) : f is an element of 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 is an element of S and t is an element of S-f, the multiplier space M (S) consisting of all sequences u such that us is an element of S whenever s is an element of 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.