Elements of Duality Theory

Abstract

In this chapter we mainly deal with relationship between a space (locally convex or Banach) and its dual space. The contents of the chapter: the general notion of duality; weak topology; polars and bipolars; the adjoint operator; Alaoglu’s theorem; \(w^*\)-convergence in a dual Banach space; the second dual space; weak convergence criteria in classical Banach spaces; total and norming sets of functionals; metrizability conditions; the Eberlein–Smulian theorem; reflexive Banach spaces.