Elements of Abstract Analysis

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.

Provides a thorough treatment of the elements of modern analysis including a systematic treatment of the set theory which analysts need to know Aimed specifically at final year undergraduates, it reviews and consolidates the mathematics learned in previous years Also serves as a book on abstract mathematics

Inhalt

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.