© 2006
Metric Spaces
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© 2006
This volume provides a complete introduction to metric space theory for undergraduates. It covers the topology of metric spaces, continuity, connectedness, compactness and product spaces, and includes results such as the Tietze-Urysohn extension theorem, Picard's theorem on ordinary differential equations, and the set of discontinuities of the pointwise limit of a sequence of continuous functions. Key features include:
a full chapter on product metric spaces, including a proof of Tychonoff’s Theorem
a wealth of examples and counter-examples from real analysis, sequence spaces and spaces of continuous functions
numerous exercises – with solutions to most of them – to test understanding.
The only prerequisite is a familiarity with the basics of real analysis: the authors take care to ensure that no prior knowledge of measure theory, Banach spaces or Hilbert spaces is assumed. The material is developed at a leisurely pace and applications of the theory are discussed throughout, making this book ideal as a classroom text for third- and fourth-year undergraduates or as a self-study resource for graduate students and researchers.
From the reviews:
"This volume provides a complete introduction to metric space theory for undergraduates. It covers the typology of metric spaces, continuity, connectedness, compactness and product spaces … . The material is developed at a leisurely pace and applications of the theory are discussed throughout, making this book ideal as a classroom text for third- and fourth-year undergraduates or as a self-study resource for graduate students and researchers." (L’Enseignement Mathematique, Vol. 51 (3-4), 2005)
"This book on metric spaces was written by authors whose main field is analysis. Therefore its focus lies on those parts of the theory of metric spaces which are mainly used in (functional) analysis. … Altogether this is an interesting book for those who will continue their studies in analysis." (H. Brandenburg, Zentralblatt Math, Vol. 1095 (21), 2006)
"This book introduces the fundamentals of analysis in metric spaces. It’s written in a very spare theorem-proof-example style; has illustrative examples and exercises; spends little time on discussion, development of intuition, or substantial applications; begins by stating that the abstract postulational method has a vital role in modern mathematics; implicitly assumes this is the way to teach mathematics. Useful resource for writing lectures? Certainly." (Donald Estep, SIAM Review, Vol. 49 (2), 2007)