Linear Normed Spaces, Linear Operators

Abstract

In this chapter structures in which also a distance is defined are studied. Additionally, these structures possess an algebraic structure, too, and are referred to as linear normed spaces. First, basic properties of linear normed spaces are investigated and fundamental differences between finite dimensional linear normed spaces and infinite-dimensional linear normed spaces are elaborated. In particular, a characterization in terms of the compactness of the unit ball is presented. Then spaces of continuous functions are studied in more detail. New notions of convergence and continuity arise and a new characterization of (relatively) compact sets in spaces of continuous functions is presented (the Arzela–Ascoli theorem). Subsequently, linear bounded operators are defined and investigated, and spaces of linear bounded operators are explored. The uniform boundedness principle is shown, and the Banach–Steinhaus theorem is proved. Finally, linear invertible operators and compact operators are studied to the degree needed for their applicability in the mathematics of computation to be presented in subsequent chapters.