Foundations of Real and Abstract Analysis

✍ Scribed by Douglas S. Bridges

• Publisher: Springer
• Year: 1997
• Tongue: English
• Leaves: 339
• Series: Graduate Texts in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

The core chapters of this volume provide a complete course on metric, normed, and Hilbert spaces, and include 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. This makes a versatile text also suited for courses on real analysis, metric spaces, abstract analysis, and modern analysis. The book begins with a comprehensive chapter providing a fast-paced course on real analysis, and is followed by an introduction to the Lebesgue integral. This provides a reference for later chapters as well as an introduction 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 such as weak solutions of the Dirichlet Problem and Pareto optimality in Mathematical Economics. Of special interest is the unique collection of nearly 750 exercises, many with guidelines for their solutions. The exercises include applications and extensions of the main propositions and theorems, results that fill in gaps in proofs or that prepare for proofs later in the book, 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 16
I Real Analysis......Page 24
1.1 The Real Number Line......Page 26
1.2 Sequences and Series......Page 35
1.3 Open and Closed Subsets of the Line......Page 50
1.4 Limits and Continuity......Page 56
1.5 Calculus......Page 68
2.1 Outer Measure and Vitali's Covering Theorem......Page 94
2.2 The Lebesgue Integral as an Antiderivative......Page 108
2.3 Measurable Sets and Functions......Page 125
II Abstract Analysis......Page 138
3.1 Metric and Topological Spaces......Page 140
3.2 Continuity, Convergence, and Completeness......Page 150
3.3 Compactness......Page 161
3.4 Connectedness......Page 173
3.5 Product Metric Spaces......Page 180
4 Analysis in Normed Linear Spaces......Page 188
4.1 Normed Linear Spaces......Page 189
4.2 Linear Mappings and Hyperplanes......Page 197
4.3 Finite–Dimensional Normed Spaces......Page 204
4.4 The L[sub(p)] Spaces......Page 209
4.5 Function Spaces......Page 219
4.6 The Theorems of Weierstrass and Stone......Page 227
4.7 Fixed Points and Differential Equations......Page 234
5.1 Inner Products......Page 248
5.2 Orthogonality and Projections......Page 252
5.3 The Dual of a Hilbert Space......Page 267
6.1 The Hahn–Banach Theorem......Page 274
6.2 Separation Theorems......Page 290
6.3 Baire's Theorem and Beyond......Page 294
A What Is a Real Number?......Page 306
B Axioms of Choice and Zorn's Lemma......Page 314
C Pareto Optimality......Page 318
References......Page 326
C......Page 332
E......Page 333
L......Page 334
O......Page 335
S......Page 336
Z......Page 337