Subsequences and rearrangements of sequences inFKspaces
Robert DeVos · Pacific Journal of Mathematics · 1976
The purpose of this paper is to study FK spaces which contain all subsequences or all rearrangements of a given sequence.Using a result of Bennett and Kalton we are able to show that if a separable FK space contains all subsequences or all rearrangements of a sequence with two or more finite cluster points, then it contains m.We are also able to show that if € p contains all rearrangements of some sequence not in / p , then it is a wedge space.This leads to proofs that if X is a solid symmetric FK space, X\i p ϊ φ, X/ s, then X^ ί p Λ for any matrix A and if in addition X is not wedge then X and ( p are not linearly homeomorphic, via a matrix, hence extending a result of Banach.1.Recently there has been a large number of papers [8], [9], [11], [13], [14] and [15] considering subsequences and rearrangements of sequences in c A and ί A .In this paper we consider these operations in an FK space setting and are able to generalize many of these results.The author would like to thank G. Bennett, F. W. Hartmann, A. K. Snyder and A. Wilansky for inspiration and many valuable conversations.Let s denote the space of all complex-valued sequences.An FK space is a vector subspace of 5 which is also a Frechet space, (complete linear metric) with continuous coordinates.A BK space is a normed FK space.Some discussion of FK spaces is given in [19].Well-known examples of BK spaces are the spaces m, c, c 0 of bounded, convergent, null sequences respectively, all with | | JC| | OC = sup|x fe |,