ABSTRACT
Among the mappings between any two vector spaces, the simplest are those which preserve the linear structure. This chapter explores the properties of those linear operators which are continuous. In fact, for linear operators, their (global) continuity turns out to be equivalent to a property that is easier to work with analytically, that of "linear boundedness". Bounded linear operators between normed spaces, and especially those between Banach spaces, have some interesting properties of paramount importance throughout Analysis and its applications. A first observation is that, for a linear operator between normed spaces, global continuity is equivalent to continuity at the origin. The chapter expounds on two deep result of Functional Analysis regarding bounded linear operators, the so-called Banach–Steinhaus Theorem and the Open Mapping Theorem. It discusses the Banach–Steinhaus Theorem. The quintessence of it may be stated informally as: any family of bounded linear mappings from a Banach space to a normed space which is pointwise bounded, actually is uniformly bounded.
