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＜E-Book＞
Real Analysis: Foundations / by Sergei Ovchinnikov
(Universitext. ISSN:21916675)

Edition	1st ed. 2021.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2021
Size	XII, 178 p. 13 illus : online resource
Authors	*Ovchinnikov, Sergei author SpringerLink (Online service)
Subjects	LCSH:Functions of real variables LCSH:Measure theory LCSH:Sequences (Mathematics) LCSH:Mathematical logic LCSH:Mathematical analysis FREE:Real Functions FREE:Measure and Integration FREE:Sequences, Series, Summability FREE:Mathematical Logic and Foundations FREE:Analysis
Notes	1 Rational Numbers -- 2 Real Numbers -- 3 Continuous Functions -- 4 Differentiation -- 5 Integration -- 6 Infinite Series -- A Natural Numbers and Integers -- B Dedekind's Construction of Real Numbers. C A Panorama of Ordered Fields This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis. Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra. Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study HTTP:URL=https://doi.org/10.1007/978-3-030-64701-8

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