Overview
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering students. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of ‘real analysis’ and ‘complex analysis’ are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.
Key Features
• Written in easy simple language, integration is described with a high degree of abstraction across the text.
• Entirely new chapter on differentiation in this edition
• Parts of certain chapters have been simplified and explained in more detail.
Table of Contents
Preface
Prologue: The Exponential Function
Chapter 1: Abstract Integration
Chapter 2: Positive Borel Measures
Chapter 3: L p -Spaces
Chapter 4: Elementary Hilbert Space Theory
Chapter 5: Examples of Banach Space Techniques Chapter 6: Complex Measures
Chapter 7: Differentiation
Chapter 8: Integration on Product Spaces
Chapter 9: Fourier Transforms
Chapter 10: Elementary Properties of Holomorphic Functions
Chapter 11: Harmonic Functions
Chapter 12: The Maximum Modulus Principle
Chapter 13: Approximation by Rational Functions
Chapter 14: Conformal Mapping
Chapter 15: Zeros of Holomorphic Functions
Chapter 16: Analytic Continuation
Chapter 17: Hp -Spaces
Chapter 18: Elementary Theory of Banach Algebras
Chapter 19: Holomorphic Fourier Transforms
Chapter 20: Uniform Approximation by Polynomials
Appendix: Hausdorff’s Maximality Theorem Notes and Comments
Bibliography
List of Special
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