Real and Convex Analysis PDF
Preview Real and Convex Analysis
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.
Thebookis built aroundmetricspaces. Inthe first three chapters,we lay thefoun-
dational 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 intro-
duction 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
importantsubjects. We gavecarefulconsiderationto what to includeand 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 ma-
terial one might find, for example, in the classic book by Rudin [Rud76] with material
onconvexityandoptimizationat a levelcommensurate,say, with the bookbyBorwein
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, optimiza-
tion plays an ever increasing role as one tries to design and analyze the best possible
“widget.”
We hope that the reader will enjoy the book and learn some important mathe-
matics.
We would like to thank the many students whom we have had the pleasure of
teaching over the years. We give a special thanks to John D’Angelo; he carefully read
a draft of the manuscript and made numerous helpful suggestions.
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