Preliminaries

The main objective of this chapter is to remind the reader of some basic notion and fundamental facts about real analysis and functional analysis required for the comprehension of the following chapters. It is assumed that readers are familiar with the concept of metric and normed spaces. The definitions of a metric space, convergence of a sequence in a metric space, completeness, compactness and the Heine Borel theorem are introduced in the first part. Here there are some well–known properties of topological concepts needed to recall. Next the notion of norm and normed spaces, equivalent norms, compactness and relatively compactness, Banach space, dual space, weak and weak* convergence are presented. Before the Hilbert spaces, inner products and inner product spaces are briefly expressed. Here the relation between the Banach and Hilbert spaces is given by some examples. Most of the theorems of that part are stated without proof since they can easily be found from any real or functional analysis book given in the reference part. Following the Fatou lemma and the Lebesgue dominated convergence theorem, the chapter ends with some important theorems based on fixed point properties on Banach spaces. For instance the Tychonoff fixed point theorem is an extension of the Schauder’s fixed point theorem and the Schauder’s theorem is an extension of the Brouwer fixed point theorem. Since the proofs of these theorems require additional knowledge, we refer the reader to the book by Papageorgiou and Winkert [1] and reference therein. In addition [2–7] may also be functional.