Duality

Abstract

AbstractFor a normed space or, more generally, a topological vector space X, we study its conjugate space X* consisting of all continuous linear functionals on X. First and foremost is the Hahn–Banach lemma, which is one of the three basic principles of linear functional analysis. It is used to prove the Hahn–Banach theorem on extending continuous linear functionals on linear subspaces, as well as many separation results. We discuss the weak and the weak* topologies. Subsequently we prove the Krein–Milman theorem on extremal points, the Stone–Weierstrass theorem, Marcinkiewicz’s interpolation theorem, and a few fixed-point theorems. We end by introducing the bounded weak* topology and proving the theorems of Dieudonne and Krein–Smulian. Finally, the chapter offer exercises to challenge the reader.