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Linear Methods Which Sum Sequences of Bounded Variation
| Content Provider | Semantic Scholar |
|---|---|
| Author | Dawson, David F. L. |
| Copyright Year | 1966 |
| Abstract | A complex sequence {zj} is said to be of bounded variation provided E Jz JP+ < 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: z < 0 } whose dual is the set of all bounded sequences. A basic tool in this study is the fact [1], [3] that a matrix (a,,) sums every sequence of bounded variation if and only if |
| Starting Page | 345 |
| Ending Page | 348 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1966-0188666-5 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0188666-5/S0002-9939-1966-0188666-5.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1966-0188666-5 |
| Volume Number | 17 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
