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Real analysis for the undergraduate. With an invitation to functional analysis. (English)
New York, NY: Springer (ISBN 978-1-4614-9637-3/hbk; 978-1-4614-9638-0/ebook). xviii, 409~p. (2014).
This textbook is intended to introduce students to the basics of real analysis. The most of the material comes from the standard undergraduate course including an axiomatic or constructive exploration of the real number system, sequences and series of numbers, limits and continuity, differentiation, sequences and series of functions, and some introductory form of integration. In addition, the reader can find there the basic measure theory and the Lebesgue integral, and a brief invitation to functional analysis. Each chapter covers a topic central to a beginning course in real analysis, the last section in each chapter introduces a topic from functional analysis which is derived in a natural way from the core chapter content. Although the book contains many advanced topics the author made them approachable for beginners without sacrificing rigor. The reader is supposed to be experienced in basic calculus (differentiation, integration, sequences and series), in basic logic and set theory, proof techniques (induction, proof by contradiction, and proof by contraposition), and basic function theory.
Reviewer: Petr Gurka (Praha)
Classification: I15
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