Linear Continuous Functionals and Dual Spaces

In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach theorem on extensions of linear functionals and the Banach-Steinhaus theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.