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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Metric Spaces - A Companion to Analysis (Paperback, 1st ed. 2022)
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Metric Spaces - A Companion to Analysis (Paperback, 1st ed. 2022)
Series: Springer Undergraduate Mathematics Series
Expected to ship within 18 - 22 working days
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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.
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