Elements of Abstract Analysis

by Mïcheïl O'Searcoid

Overview

In nature's infinite book ofsecrecy A little I can read. Antony and Cleopatra, l. ii. This is a book about a few elementary concepts of analysis and the mathe­ matical structures which enfold them. It is more concerned with the interplay amongst these concepts than with their many applications. The book is self-contained; in the first chapter, after acknowledging the fundamental role ofmathematical logic, wepresent seven axioms of Set Theory; everything else is developed from these axioms. It would therefore be true, if misleading, to say that the reader requires no prior knowledge of mathematics. In reality, the reader we have in mind has that level of sophistication achieved in about three years of undergraduate study of mathematics and is already well acquainted with most of the structures discussed-rings, linear spaces, metric spaces, and soon-and with many of the principal analytical concepts­ convergence, connectedness, continuity,compactness and completeness. Indeed, it is only after gaining familiarity with these concepts and their applications that it is possible to appreciate their place within a broad framework of set­ based mathematics and to consolidate an understanding of them in such a framework. To aid in these pursuits, wepresent our reader with things familiar and things new side by side in most parts of the book-and we sometimes adopt an unusual perspective. That this is not an analysis textbook is clear from its many omissions.

Product Details

ISBN-13:	9781852334246
Publisher:	Springer London
Publication date:	12/06/2001
Series:	Springer Undergraduate Mathematics Series
Edition description:	2002
Pages:	300
Product dimensions:	7.01(w) x 9.25(h) x 0.38(d)

Table of Contents

1. Sets.- 1.1 Set Theory.- 1.2 Relations and Functions.- 1.3 Ordered Sets.- 1.4 Ordinals.- 1.5 The Axiom of Choice.- 2. Counting.- 2.1 Counting Numbers.- 2.2 Cardinality.- 2.3 Enumeration.- 2.4 Cardinality of Unions and Products.- 3. Algebraic Structure.- 3.1 Elementary Algebraic Structures.- 3.2 Vector Spaces.- 3.3 Algebras.- 3.4 Preservation of Algebraic Structure.- 4. Analytic Structure.- 4.1 Ordered Algebraic Structure.- 4.2 Number Systems.- 4.3 Real and Complex Functions.- 4.4 Inequalities.- 5. Linear Structure.- 5.1 Linear Spaces and Algebras.- 5.2 Linear Shapes.- 5.3 Linear Functionals.- 6. Geometric Structure.- 6.1 Semimetrics and Metrics.- 6.2 Seminorms and Norms.- 6.3 Sesquilinear Forms and Inner Products.- 7. Topological Structure.- 7.1 Topologies.- 7.2 Neighbourhoods.- 7.3 Cardinality and Topology.- 7.4 Separation.- 8. Continuity and Openness.- 8.1 Preservation of Topological Structure.- 8.2 Topologies Denned by Functions.- 8.3 Derived Topological Spaces.- 8.4 Topologies on Linear Spaces.- 9. Connectedness.- 9.1 Connected Spaces.- 9.2 Pathwise Connectedness.- 10. Convergence.- 10.1 Filters.- 10.2 Limits.- 11. Compactness.- 11.1 Compact Topological Spaces.- 11.2 Compact Hausdorff Spaces.- 11.3 Local Compactness.- 12. Completeness.- 12.1 Complete Metric Spaces.- 12.2 Banach Spaces.- 12.3 Hilbert Spaces.- 12.4 Banach Algebras.- Solutions.

Editorial Reviews

From the reviews of the first edition:

BULLETIN OF MATHEMATICS BOOKS

"…an interesting mix of algebra, topology and functional analysis, with chapter titles like Algebraic Structure, Linear Structure, Geometric Structure and Topological Structure."

"Michael O’Searcoid’s book ‘Element of Abstract Analysis’ is a book that deals with the fundamental principles of mathematics in a broad framework of set-theoretic approach. The book is written with a deep insight. … The contents are conceptually useful to students. … The book is a nice readable treat for every student of mathematics and is highly recommended for one semester course at M.Sc. level." (Madhu Tiwari, Journal of the Indian Academy of Mathematics, Vol. 25 (1), 2003)

"This book covers the elements of abstract analysis, treated mostly within the context of topological spaces. … This is a carefully written book, starting from set-theoretic axioms, and containing about 180 problems and exercises with hints and solutions. It is an ideal source of information for independent student study. There is a comprehensive index." (European Mathematical Society Newsletter, September, 2002)

"The book presents some concepts of analysis and the mathematical structures which enfold them. … the reviewer recommends the book to everyone interested in analysis, especially those scholars interested in studying more advanced areas like Functional Analysis, or Operator Theory." (Catalin Badea, Zentralblatt MATH, Vol. 999 (24), 2002)

"The book is written specifically for final-year undergraduate students who should already be familiar with most of the mathematical structures discussed … . It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics. This book is a rigorous, self-contained introduction to functional analysis that will also serve as a text onabstract mathematics." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)

"This is a book about a few elementary concepts of analysis and the mathematical structures which enfold them. … The book is written specifically for final-year and undergraduate students who should already be familiar with most of the mathematical structures discussed … and with many of the principal analytical concepts … . It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics." (Zentralblatt für Didaktik der Mathematik, February, 2002)

From the Publisher