Introduction to the Analysis of Metric Spaces

Introduction to the Analysis of Metric Spaces

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This is an introduction to the analysis of metric and normed linear spaces for undergraduate students in mathematics. Assuming a basic knowledge of real analysis and linear algebra, the student is exposed to the axiomatic method in analysis and is shown its power in exploiting the structure of fundamental analysis, which underlies a variety of applications. An example is the link between normed linear spaces and linear algebra; finite dimensional spaces are discussed early. The treatment progresses from the concrete to the abstract: thus metric spaces are studied in some detail before general topology is begun, though topological properties of metric spaces are explored in the book. Graded exercises are provided at the end of each section; in each set the earlier exercises are designed to assist in the detection of the structural properties in concrete examples while the later ones are more conceptually more

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Table of contents

Preface; Part I. Metric Spaces and Normed Linear Spaces: 1. Definitions and examples; 2. Balls and boundedness; Part II. Limit Processes: 3. Convergence and completeness; 4. Cluster points and closure; 5. Application: Banach's fixed point theorem; Part III. Continuity: 6. Continuity in metric spaces; 7. Continuous linear mappings; Part IV. Compactness: 8. Sequential compactness in metric spaces; 9. Continuous functions on compact metric spaces; Part V. The Metric Topology: 10. The topological analysis of metric spaces; Appendices; Index of notation; more