On Relationships Amongst Certain Spaces Of Sequences In An Arbitrary Banach Space

Abstract

1. Introduction. Let X be a Banach space ( B -space). A sequence {s(i)} in X is unconditionally summable if and only if every rearrangement of the series Σ i s(i) is convergent. The set of unconditionally summable sequences in X will be written as U(X). In this paper several classes of summable sequences in X will be compared with one another. Each class to be considered is identical with U(X) when X has finite dimension.