On unconditionally converging series and biorthogonal systems in a Banach space

Abstract

Our main result is as follows: Let B be a Banach space containing no subspace isomorphic (linearly homeomorphic) to Zoo, and let {(b n ,βn)} be a biorthogonal sequence in B such that (β n ) is total.If xeB then Σn=i βn(x)b n converges unconditionally to x if and only if for every sequence (a n ) of O's and Γs there exists y e B with β n (y) = a n β n (x) for all n.This theorem improves previous results of Kadec and Pelczynski.Similar results are obtained in the context of biorthogonal decompositions of a Banach space into separable subspaces.

Locations

• Pacific Journal of Mathematics - View - PDF