Isaac Scientific Publishing

Advances in Analysis

Weak Compactness in Abstract Duality Pairs

Download PDF (692.1 KB) PP. 135 - 142 Pub. Date: March 9, 2017

DOI: 10.22606/aan.2017.22008

Author(s)

  • Charles Swartz*
    Department of Mathematics, New Mexico State University, Las Cruces, NM 88003, USA

Abstract

Let E, F be (real) vector spaces and G a topological vector space and assume that there is a bilinear map b : E × F → G . We call E, F,G an abstract triple (abstract duality pair with respect to G) and denote it by (E, F : G). We write b(x, y) = x · y for x 2 E, y 2 F. The weakest topology on E such that all of the linear maps x→ x · y from E into G are continuous for y 2 F is denoted by w(E, F). We study sequential compactness and sequential completeness for this topology when E is a space of vector valued, bounded, finitely additive set functions or the space of Bochner or Pettis integrable functions, F is a space of bounded measurable functions, G is a Banach space and the bilinear map is defined via an integral. We also consider vector valued sequence spaces.

Keywords

Weak compact, vector valued set functions, Bochner integral, Pettis integral, sequence spaces.

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