Chapter 1: Abstract Duality Pairs or Abstract Triples

Abstract:

The notion of abstract duality pairs or abstract triples is a generalization of the notion of dual pairs of vector spaces which is utilized in the development of locally convex spaces. A pair of (real) vector spaces E, F is a dual pair if there exists a bilinear map

which separates points. The idea of abstract duality pairs or abstract triples is to replace the scalar field ℝ by a Hausdorff Abelian topological group G and to replace the vector spaces E, F by arbitrary sets; in most applications the sets E, F have additional structures. A theory of abstract triples was initially developed by Professor Ronglu Li and Charles Swartz when Professor Li was a visiting scholar in the mathematics department of New Mexico State University from 1988–1990. The development continued after Professor Li returned to his home university, Harbin Institute of Technology, and Professor Li later engaged Professor Min-Hyung Cho of Kum-Oh National Institute of Technology in Kumi, Korea, to join in the development. This resulted in a series of notes on the subject but the notes were never published although the idea of abstract triples has appeared in various places in the literature. In this text we will present the results of the original notes as well as a number of further developments and applications of the theory. The idea has found applications in a number of different areas of analysis. In particular, we give applications to general versions of the Orlicz–Pettis Theorem on subseries convergence of series, various topics in the theory of vector valued measures and vector valued integrals, sequence spaces, multiplier convergent series, the Uniform Boundedness Principle, the Banach–Steinhaus Theorem, the Mazur–Orlicz Theorem on bilinear operators and weak compactness…