This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. 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: the basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. Proofs of theorems presented in the book are concise and complete. Challenging and through provoking exercise at the end of each paper. Table of content: preface prologue: the exponential function chapter 1: abstract integration chapter 2: positive borel measures chapter 3: lp-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
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