Çınlar Erhan, Vanderbei Robert J. Real and Convex Analysis

New York: Springer Science+Business Media, 2013. — 161 p. — (Undergraduate texts in mathematics). — ISBN: 978-1-4614-5256-0, 978-1-4614-5257-7.This book is intended to serve as a first course in analysis for scientists and engineers. It can be used either at the advanced undergraduate level or as part of the curriculum in a graduate program. We have taught from preliminary drafts of the book for several years.
The book is built around metric spaces. In the first three chapters, we lay the foundational material. We cover the all-important four Cs: convergence, completeness, compactness, and continuity. We have organized the material to be as simple and as logical as possible.
In subsequent chapters, we use the basic tools of analysis to give a brief introduction to closely related topics such as differential and integral equations, convex analysis, and measure theory. The book is short and yet covers in some depth the most important subjects. We gave careful consideration to what to include and what to leave out. In all such considerations, we asked ourselves whether the material would be of direct and immediate use to scientists and engineers. Our philosophy is if in doubt, do without.
What makes this book different? We pull together some of the foundational material one might find, for example, in the classic book by Rudin [Rud76] with material on convexity and optimization at a level commensurate, say, with the book by Borwein and Lewis [BL06] and with a completely modern treatment of the basics of measure theory. The importance of measure theory has increased over the years as stochastic modeling has become more central to all aspects of analysis. Similarly, optimization plays an ever increasing role as one tries to design and analyze the best possible widget.