ABSTRACT
Chapter 1 provides a survey of useful fundamental notations and concepts such as partially ordered and directed sets, and Zorn's lemma, as well as important inequalities, such as the Hölder, Minkowski and Jensen inequalities. It also contains a review of fundamental well–known concepts and results from linear algebra concerning linear spaces, linear maps, which are essential throughout the book. Furthermore, it revises the concepts of semimetric, seminormed and topological spaces, open and closed sets and neighborhoods, continuity of functions between topological spaces, bases and subbases of topological spaces, and separation axioms. Finally, detailed studies are dedicated to net convergence, subnets and compactness in topological spaces, and the proofs are given for those results which the authors regard as being less familiar. Preference was given to net convergence over filter convergence, since net convergence seems to be the more natural extension to topological spaces of convergence of sequences in semimetric, metric and normed spaces.
