Elements of Abstract Analysis - Searcóid, Mícheál Ó (Springer London, 2002)
In nature's infinite book of secrecy A little I can read. Antony and Cleopatra, 1. ii. This is a book about a few elementary concepts of analysis and the mathematical 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 of mathematical logic, we present 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 setbased mathematics and to consolidate an understanding of them in such a framework. To aid in these pursuits, we present 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