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This book, which is the first of two volumes, presents, in a unique way, some of the most relevant research tools of modern analysis. This work empowers young researchers with all the necessary techniques to explore the various subfields of this broad subject, and introduces relevant frameworks where these tools can be immediately deployed.
Volume I starts with the foundations of modern analysis. The first three chapters are devoted to topology, measure theory, and functional analysis. Chapter 4 offers a comprehensive analysis of the main function spaces, while Chapter 5 covers more concrete subjects, like multivariate analysis, which are closely related to applications and more difficult to find in compact form. Chapter 6 deals with smooth and non-smooth calculus of functions; Chapter 7 introduces certain important classes of nonlinear operators; and Chapter 8 complements the previous three chapters with topics of variational analysis.
Eachchapter of this volume finishes with a list of problems - handy for understanding and self-study - and historical notes that give the reader a more vivid picture of how the theory developed. Volume II consists of various applications using the tools and techniques developed in this volume.
By offering a clear and wide picture of the tools and applications of modern analysis, this work can be of great benefit not only to mature graduate students seeking topics for research, but also to experienced researchers with an interest in this vast and rich field of mathematics.
List of contents
Volume I - Theory: - Topology.- Measure Theory.- Banach Space Theory.- Function Spaces.- Multivalued Analysis.- Smooth and Nonsmooth Calculus.- Nonlinear Operators.- Variational Analysis.- References.
About the author
¿Shouchuan Hu is a Distinguished Professor at Missouri State University, USA. He holds a PhD in Mathematics from the University of Texas at Arlington. Dr. Hu is Director of the AIMS - American Institute of Mathematical Sciences and Editor-in-Chief of the "Discrete and Continuous Dynamical Systems" journal. He co-authored, along with Dr. Papageorgiou, the two-volume set "Handbook of Multivalued Analysis" (1997), published by Springer.
Nikolaos S. Papageorgiou is a Professor at the National Technical University of Athens, Greece. He holds a PhD in Applied Mathematics (1983) from Harvard University, USA, and degrees in Mathematics and Electrical Engineering, both from Massachusetts Institute of Technology - MIT. Dr. Papageorgiou has co-authored over a dozen books, including "Exercises in Analysis" (2016) and "Nonlinear Analysis - Theory and Methods" (2019), both published by Springer.
Report
"Each chapter closes with interesting historical remarks and further bibliographical hints, followed by a collection of problems and exercises, ranging from the completely obvious to the really challenging. This makes the book a valuable source for postdoctoral students, reseachers, and anybody else who wants to get an impression of classical fields of higher analysis, with particular emphasis on nonlinear problems. The book is well written, and it contains a wealth of material." (Jürgen Appell, zbMATH 1514.46001, 2023)
"I recommend this text as a standalone text for any student who is trying to learn this material for the first time. ... I conjecture that, once paired, the two-volume set will make an excellent tool for researchers looking to quickly gain insight into the tools used in modern analytical research." (John Ross, MAA Reviews, June 17, 2023)
