Real and Convex Analysis

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. Table of Contents Cover Real and Convex Analysis ISBN 9781461452560 ISBN 9781461452577 Preface Contents Notation and Usage Chapter 1. Sets and Functions A. Sets B. Functions and Sequences C. Countability D. On the Real Line E. Series Chapter 2. Metric Spaces A. Euclidean Spaces B. Metrics C. Open and Closed Sets D. Convergence E. Completeness F. Compactness Chapter 3. Functions on Metric Spaces A. Continuous Mappings B. Compactness and Uniform Continuity C. Sequences of Functions D. Spaces of Continuous Functions Chapter 4. Differential and Integral Equations A. Contraction Mappings B. Systems of Linear Equations C. Integral Equations D. Differential Equations Chapter 5. Convexity A. Convex Sets and Convex Functions B. Projections C. Supporting Hyperplane Theorem D. Legendre Transform E. Infimal Convolution Chapter 6. Convex Optimization A. Primal and Dual Problems B. Linear Programming and Polyhedra C. Lagrangians D. Saddle Points Chapter 7. Measure and Integration A. Algebras B. Measurable Spaces and Functions C. Measures D. Integration E. Transforms and Indefinite Integrals F. Kernels and Product Spaces Further Reading Bibliography Index