Unconditional and shrinking bases in locally convex spaces

Leon Jarome Weill

Vol. 29 (1969), No. 2, 467–483

DOI: 10.2140/pjm.1969.29.467

Abstract

Let E be a locally convex space with an unconditional Schauder basis {x_k} and let {f_k} be the sequence of coefficient functionals biorthogonal to {x_k}. Owing to works of R. C. James and S. Karlin it is known that if E is a Banach space then each of the three conditions which follow is necessary and sufficient for {f_k} to be a basis for E^∗ in the strong (norm) topology.

1. E has no subspace topologically isomorphic to the space l¹.
2. E^∗ is separable in the strong topology.
3. E^∗ is weakly (w(E^∗,E^∗∗)) sequentially complete.

The primary purpose of this paper is to show that in certain spaces which are more general than Frechet spaces and hence than Banach spaces, each of the above three conditions is necessary and sufficient for

(0) {f_k} is a strong basis for E^∗.

Mathematical Subject Classification

Primary: 46.01

Milestones

Received: 3 April 1967

Published: 1 May 1969