Metric spaces

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.

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.

"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." "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."--Jacket This book treats material concerning Metric Spaces, which is crucial for any advanced level course in analysis. Usually this is taught in the latter years of an undergraduate course. Often books on functional analysis have to be used, because there are so few books specifically on metric spaces ? a gap that needs to be filled in the literature. It is a comprehensive introduction, with applications, containing a large number if worked examples and exercises for real and complex analysis, sequence spaces and the space of continuous functions. Each concept is followed by a large number of examples, some satisfying the requirements and others not, so that the concept is well understood. One of the first books to be dedicated specifically to metric spaces Full of worked examples, to get complex ideas across more easily In many branches of mathematics, it is convenient to have available a notion of distance between elements of an abstract set.