Linear Methods Which Sum Sequences of Bounded Variation

A complex sequence {zp} is said to be of bounded variation provided 2|zp —zP+i\ < oo. In this paper we show that a matrix which sums every sequence of bounded variation also sums a convergent sequence not of bounded variation (Theorem 1). Indeed, if M is a countable set of matrices, each of which sums every sequence of bounded variation, then there is a convergent sequence not of bounded variation which every matrix in M sums (Theorem 2). Our proofs are by direct construction. We are indebted to the referee for the following observation: Theorem 1 follows from a rather inaccessible result of Mazur-Orlicz-Zeller (see p. 125 of [4] and p. 256 of [5]) to the effect that the set of all convergent sequences which a matrix sums, as an FK space, has a separable dual space, while the space of sequences of bounded variation does not, since its maximal subspace of null sequences is equivalent to {z: ]C|Z«| < °° } whose dual is the set of all bounded sequences. A basic tool in this study is the fact [l], [3] that a matrix iaP9) sums every sequence of bounded variation if and only if