Bounded Linear Maps

Abstract

Having described the basic framework of a normed space in the previous chapter, we study the continuity of linear maps between normed spaces in this chapter. The notion of the operator norm of a continuous linear map is important in this context. We give many examples of continuous linear maps which include matrix transformations and Fredholm integral maps, and attempt to find their operator norms. Four major results are proved in the second and the third section: the uniform boundedness principle, the closed graph theorem, the bounded inverse theorem and the open mapping theorem. These are easily deduced from a theorem of Zabreiko which states that a countably subadditive seminorm on a Banach space is continuous. We give several applications of these major results. In the last section of this chapter, we introduce compact linear maps. They provide a useful generalization of finite rank continuous maps.