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.
Each chapter 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.
