Elements of Abstract Analysis

✍ Scribed by Mícheál Ó Searcóid

• Publisher: Springer
• Year: 2002
• Tongue: English
• Leaves: 310
• Series: Springer Undergraduate Mathematics Series
• Edition: 1
• Category: Library

✦ Synopsis

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 ofthe 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.

✦ Table of Contents

Front Matter....Pages i-xii
Sets....Pages 1-40
Counting....Pages 41-56
Algebraic Structure....Pages 57-90
Analytic Structure....Pages 91-114
Linear Structure....Pages 115-132
Geometric Structure....Pages 133-158
Topological Structure....Pages 159-176
Continuity and Openness....Pages 177-206
Connectedness....Pages 207-214
Convergence....Pages 215-230
Compactness....Pages 231-244
Completeness....Pages 245-268
Back Matter....Pages 269-300

✦ Subjects

Analysis