Some Remarks to the Preceding Chapters. Normed Space, Banach Space

Abstract

The main objective of the preceding chapters was the development of the theory of the Hilbert space which will be encountered in the sequel in most different forms. Therefore, we have started from the concept of a linear set and introduced the inner product, the norm, and the distance, on this set. A number of concepts and, also, of results concerning metric spaces were presented for this linear case only. In this chapter, we will draw the reader’s attention — at least briefly — to problems of which of these concepts (or results) can or cannot be extended to nonlinear spaces. The reader interested in proceeding more rapidly to the study of variational methods, as discussed in the following chapters, may omit Chap. 7 for the present.