Real and Convex Analysis

Description

This book offers a first course in analysis for scientists and engineers. It can be used at the advanced undergraduate level or as part of the curriculum in a graduate program. The book is built around metric spaces. In the first three chapters, the authors lay the foundational material and cover the all-important “four-C’s”: convergence, completeness, compactness, and continuity. In subsequent chapters, the basic tools of analysis are used to give brief introductions to differential and integral equations, convex analysis, and measure theory. The treatment is modern and aesthetically pleasing. It lays the groundwork for the needs of classical fields as well as the important new fields of optimization and probability theory. Cover......Page 3 Preface......Page 5 Contents......Page 7 Notation and Usage......Page 9 A. Sets......Page 11 B. Functions and Sequences......Page 14 C. Countability......Page 16 D. On the Real Line......Page 19 E. Series......Page 24 A. Euclidean Spaces......Page 32 B. Metrics......Page 34 C. Open and Closed Sets......Page 38 D. Convergence......Page 44 E. Completeness......Page 46 F. Compactness......Page 50 A. Continuous Mappings......Page 55 B. Compactness and Uniform Continuity......Page 60 C. Sequences of Functions......Page 64 D. Spaces of Continuous Functions......Page 67 A. Contraction Mappings......Page 72 B. Systems of Linear Equations......Page 77 C. Integral Equations......Page 80 D. Differential Equations......Page 87 A. Convex Sets and Convex Functions......Page 92 B. Projections......Page 95 C. Supporting Hyperplane Theorem......Page 98 D. Legendre Transform......Page 99 E. Infimal Convolution......Page 105 A. Primal and Dual Problems......Page 107 B. Linear Programming and Polyhedra......Page 112 C. Lagrangians......Page 114 D. Saddle Points......Page 115 A. Algebras......Page 120 B. Measurable Spaces and Functions......Page 123 C. Measures......Page 130 D. Integration......Page 136 E. Transforms and Indefinite Integrals......Page 146 F. Kernels and Product Spaces......Page 151 Further Reading......Page 159 Bibliography......Page 160 Index......Page 161