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Metric Spaces

A Companion to Analysis

  • Textbook
  • © 2022

Overview

Authors:
  1. Robert Magnus

    1. Faculty of Physical Sciences, University of Iceland, Reykjavik, Iceland

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  • Provides a lucid and clear exposition which includes additional motivation and explanation for delicate points
  • Presents metric spaces as a tool for advanced analysis, topology and related subjects
  • Includes many exercises with hints

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xix
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  2. Metric Spaces

    • Robert Magnus
    Pages 1-27

  3. Basic Theory of Metric Spaces

    • Robert Magnus
    Pages 29-85

  4. Completeness of the Classical Spaces

    • Robert Magnus
    Pages 87-123

  5. Compact Spaces

    • Robert Magnus
    Pages 125-162

  6. Separable Spaces

    • Robert Magnus
    Pages 163-186

  7. Properties of Complete Spaces

    • Robert Magnus
    Pages 187-218

  8. Connected Spaces

    • Robert Magnus
    Pages 219-237

  9. Back Matter

    Pages 239-244
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Keywords

About this book

This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.

The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves.


This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.

Reviews

“I would enthusiastically recommend this book for a student who has already taken a basic real analysis course … . I think it is a real winner. It is very approachable and well-paced … its exercises are well thought out; and through its excursions and exposition, it gives the reader a solid foundation in metric space theory, with an understanding of where this theory sits within the broader fields of topology and analysis.” (John Ross, MAA Reviews, February 19, 2023)

Authors and Affiliations

  • Faculty of Physical Sciences, University of Iceland, Reykjavik, Iceland

    Robert Magnus

About the author

Robert Magnus was born in the UK and studied mathematics at the Universities of Cambridge and Sussex. He has taught nearly all subjects associated with analysis and has published papers in the areas of bifurcation theory, catastrophe theory, analytic operator functions and nonlinear partial differential equations. Since 1977 he has lived and worked in Iceland.

Bibliographic Information

  • Book Title: Metric Spaces

  • Book Subtitle: A Companion to Analysis

  • Authors: Robert Magnus

  • Series Title: Springer Undergraduate Mathematics Series

  • DOI: https://doi.org/10.1007/978-3-030-94946-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Softcover ISBN: 978-3-030-94945-7Published: 17 March 2022

  • eBook ISBN: 978-3-030-94946-4Published: 16 March 2022

  • Series ISSN: 1615-2085

  • Series E-ISSN: 2197-4144

  • Edition Number: 1

  • Number of Pages: XIX, 244

  • Number of Illustrations: 10 b/w illustrations, 1 illustrations in colour

  • Topics: Analysis, Topology, Geometry

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