On unconditionally converging series and biorthogonal systems in a Banach space

Gregory Frank Bachelis and Haskell Paul Rosenthal

Vol. 37 (1971), No. 1, 1–5

DOI: 10.2140/pjm.1971.37.1

Abstract

Our main result is as follows: Let B be a Banach space containing no subspace isomorphic (linearly homeomorphic) to l_∞, and let {(b_n,β_n)} be a biorthogonal sequence in B such that (β_n) is total. If x ∈ B then ∑ _n=1^∞β_n(x)b_n converges unconditionally to x if and only if for every sequence (a_n) of 0’s and l’s there exists y ∈ B with β_n(y) = a_nβ_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.

Mathematical Subject Classification 2000

Primary: 46B15

Milestones

Received: 5 February 1970

Revised: 17 July 1970

Published: 1 April 1971