DUAL SPACES

This lecture deals primarily (though not exclusively) with the most important
class of topological vector spaces, namely, the locally convex ones. The
highlights, from the theoretical as well as the applied standpoints, are
(a) the Hahn-Banach theorems (assuring a supply of continuous linear functionals
that is adequate for a highly developed duality theory), (b) the
Banach-Alaoglu compactness theorem in dual spaces, and (c) the KreinMilman
theorem about extreme points. Applications to various problems in
analysis.This part deals primarily (though not exclusively) with the most important
class of topological vector spaces, namely, the locally convex ones. The
highlights, from the theoretical as well as the applied standpoints, are
(a) the Hahn-Banach theorems (assuring a supply of continuous linear functionals
that is adequate for a highly developed duality theory), (b) the
Banach-Alaoglu compactness theorem in dual spaces, and (c) the KreinMilman
theorem about extreme points. Applications to various problems in
analysis are postponed to Chapter 5.