Synopses & Reviews
The focus of this modern graduate text in real analysis is to prepare the potential researcher to a rigorous "way of thinking" in applied mathematics and partial differential equations. The book will provide excellent foundations and serve as a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. All the core topics of the subject are covered, from a basic introduction to functional analysis, to measure theory, integration and weak differentiation of functions, and in a presentation that is hands-on, with little or no unnecessary abstractions. Additional features: * Carefully chosen topics, some not touched upon elsewhere: fine properties of integrable functions as they arise in applied mathematics and PDEs - Radon measures, the Lebesgue Theorem for general Radon measures, the Besicovitch covering Theorem, the Rademacher Theorem; topics in Marcinkiewicz integrals, functions of bounded variation, Legendre transform and the characterization of compact subset of some metric function spaces and in particular of Lp spaces * Constructive presentation of the Stone-Weierstrass Theorem * More specialized chapters (8-10) cover topics often absent from classical introductiory texts in analysis: maximal functions and weak Lp spaces, the Calderón-Zygmund decomposition, functions of bounded mean oscillation, the Stein-Fefferman Theorem, the Marcinkiewicz Interpolation Theorem, potential theory, rearrangements, estimations of Riesz potentials including limiting cases * Provides a self-sufficient introduction to Sobolev Spaces, Morrey Spaces and Poincaré inequalities as the backbone of PDEs and as an essential environment to develop modern and current analysis * Comprehensive index This clear, user-friendly exposition of real analysis covers a great deal of territory in a concise fashion, with sufficient motivation and examples throughout. A number of excellent problems, as well as some remarkable features of the exercises, occur at the end of every chapter, which point to additional theorems and results. Stimulating open problems are proposed to engage students in the classroom or in a self-study setting.
"The book serves as a solid building block for research in analysis, PDEs, the calculus of variations, probability theory and approximation theory; it provides an excellent foundation of real analysis."
This book covers the core mathematical topics of analysis and will attract the reader with a more applied view. The focus is to prepare the potential researcher to a "way of thinking" in applied mathematics and partial differential equations. The exposition is hands-on and accommodating to this group with little or no unnecessary abstractions.
This book is a self-contained introduction to real analysis assuming only basic notions on limits of sequences in ]RN, manipulations of series, their convergence criteria, advanced differential calculus, and basic algebra of sets. The passage from the setting in ]RN to abstract spaces and their topologies is gradual. Continuous reference is made to the ]RN setting, where most of the basic concepts originated. The first seven chapters contain material forming the backbone of a basic training in real analysis. The remaining two chapters are more topical, relating to maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions. Even though the layout of the book is theoretical, the entire book and the last chapters in particular concern applications of mathematical analysis to models of physical phenomena through partial differential equations. The preliminaries contain a review of the notions of countable sets and related examples. We introduce some special sets, such as the Cantor set and its variants, and examine their structure. These sets will be a reference point for a number of examples and counterexamples in measure theory (Chapter II) and in the Lebesgue differentiability theory of absolute continuous functions (Chapter IV). This initial chapter also contains a brief collection of the various notions of ordering, the Hausdorff maximal principle, Zorn's lemma, the well-ordering principle, and their fundamental connections.
This graduate text in real analysis is a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. It covers all the core topics, such as a basic introduction to functional analysis, and it discusses other topics often not addressed including Radon measures, the Besicovitch covering Theorem, the Rademacher theorem, and a constructive presentation of the Stone-Weierstrass Theoroem.
Table of Contents
Preface * Preliminaries * Topologies and Metric Spaces * Measuring Sets * The Lebesgue Integral * Topics on Measurable Functions of Real Variables * The L^p Spaces * Banach Spaces * Spaces of Continuous Functions, Distributions, and Weak Derivitives * Topics on Integrable Functions of Real Variables * Embedding of W ^1,p (E) into L^q (E) * References * Index