A dual geometric characterization of Banach spaces not containing l₁

Elias Saab and Paulette Saab

Vol. 105 (1983), No. 2, 415–425

DOI: 10.2140/pjm.1983.105.415

Abstract

It is shown that a Banach space E does not contain a copy of l₁ if and only if every bounded subset of E^∗ is w^∗-dentable in (E^∗,σ(E^∗,E^∗∗)). The notion of w^∗-scalarly dentable sets in dual Banach space is introduced and it is proved that a Banach space E does not contain a copy of l₁ if and only if every bounded set in E^∗ is w^∗-scalarly dentable. Finally, a point of continuity criterion that characterizes Asplund operators and those operators that factor through Banach spaces not containing copies of l₁, is given.

Mathematical Subject Classification 2000

Primary: 46B20

Milestones

Received: 23 September 1981

Published: 1 April 1983