Foundations of real and abstract analysis

✍ Scribed by Bridges D.

• Publisher: Springer
• Year: 1998
• Tongue: English
• Leaves: 340
• Series: Graduate texts in mathematics -174
• Category: Library

✦ Synopsis

A complete course on metric, normed, and Hilbert spaces, including many results and exercises seldom found in texts on analysis at this level. The author covers an unusually wide range of material in a clear and concise format, including elementary real analysis, Lebesgue integration on R, and an introduction to functional analysis. The book begins with a fast-paced course on real analysis, followed by an introduction to the Lebesgue integral. This provides a reference for later chapters as well as a preparation for students with only the typical sequence of undergraduate calculus courses as prerequisites. Other features include a chapter introducing functional analysis, the Hahn-Banach theorem and duality, separation theorems, the Baire Category Theorem, the Open Mapping Theorem and their consequences, and unusual applications. Of special interest are the 750 exercises, many with guidelines for their solutions, applications and extensions of the main propositions and theorems, pointers to new branches of the subject, and difficult challenges for the very best students.

✦ Table of Contents

Contents......Page 14
Preface......Page 10
Introduction......Page 17
I Real Analysis......Page 25
1.1 The Real Number Line......Page 27
1.2 Sequences and Series......Page 36
1.3 Open and Closed Subsets of the Line......Page 51
1.4 Limits and Continuity......Page 57
1.5 Calculus......Page 69
2.1 Outer Measure and Vitali's Covering Theorem......Page 95
2.2 The Lebesgue Integral as an Antiderivative......Page 109
2.3 Measurable Sets and Functions......Page 126
II Abstract Analysis......Page 139
3.1 Metric and Topological Spaces......Page 141
3.2 Continuity, Convergence, and Completeness......Page 151
3.3 Compactness......Page 162
3.4 Connectedness......Page 174
3.5 Product Metric Spaces......Page 181
4 Analysis in Normed Linear Spaces......Page 189
4.1 Normed Linear Spaces......Page 190
4.2 Linear Mappings and Hyperplanes......Page 198
4.3 Finite–Dimensional Normed Spaces......Page 205
4.4 The L[sub(p)] Spaces......Page 210
4.5 Function Spaces......Page 220
4.6 The Theorems of Weierstrass and Stone......Page 228
4.7 Fixed Points and Differential Equations......Page 235
5.1 Inner Products......Page 249
5.2 Orthogonality and Projections......Page 253
5.3 The Dual of a Hilbert Space......Page 268
6.1 The Hahn–Banach Theorem......Page 275
6.2 Separation Theorems......Page 291
6.3 Baire's Theorem and Beyond......Page 295
Appendix A: What Is a Real Number?......Page 307
Appendix B: Axioms of Choice and Zorn's Lemma......Page 315
Appendix C: Pareto Optimality......Page 319
References......Page 327
Index......Page 333