ABSTRACT
Chapter 4 concentrates on the studies of Banach spaces. It contains the most important examples of Banach spaces and results on the bounded linear operators between normed and Banach spaces. Highlights are the fundamental Hahn–Banach extension theorem and several of its corollary theorems which are vitally important in the proof of many results in functional analysis. Furthermore, the versions in Banach spaces are given of the open mapping and closed graph theorems, Banach’s theorem of the bounded inverse, and the Banach–Steinhaus theorem. Two important applications of the closed graph theorem are the criterion for a closed subspace in a Banach space to have a topological complement, and the Eni–Karauš theorem. Further applications of the Hahn–Banach and Banach–Steinhaus theorems are the classical representation theorems for the continuous linear functionals on the classical sequence spaces, including the representation theorem for the space of bounded real sequences, and the Riesz representation theorem for the continuous linear functionals on the space of continuous real functions on the unit interval. The other topics focus on the reflexivity of spaces, the studies of adjoint operators, quotient spaces, Cauchy nets, the equivalence of norms, compactness and the Riesz lemma, compact operators and operators with closed range.
