Mathematical Analysis III

This module is an introduction to analysis in the setting of metric spaces. There are at least two advantages by adopting this slightly abstract point of view. First of all, it helps to crystallize fundamental concepts and elucidate the roles they play in the theory. Secondly, it provides a unified framework for applications of the results and techniques of mathematical analysis. This module will cover the basic theory of metric spaces and sample applications to other areas of mathematics. It is highly recommended to students majoring in pure mathematics and to those who are interested in applied mathematics with an analytical flavour. Major topics: Euclidean spaces, inner product and Euclidean norm. Metric spaces: definition, examples. Topological concepts: open sets and closed sets, subspaces, density and separability. Convergence of sequences, completeness, nowhere dense sets, Baires category theorem and applications. Continuity of functions and uniform continuity. Compactness: open covers, Heine-Borel Theorem, extreme value theorem. Equivalences of compactness, sequential compactness, and completeness and total boundeness. Connectedness, characterizations of subintervals of the real line, intermediate value theorem, path-connectedness. Contraction mappings, Banachs fixed point theorem and applications. Function spaces: pointwise and uniform convergence for sequences and series of functions, Weierstrass M-test, boundedness and equicontinuity, Arzela-Ascoli Theorem. Weierstrass Approximation Theorem and applications.

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