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Real Analysis

  • Emmanuele DiBenedetto

Part of the Birkhäuser Advanced Texts book series (BAT)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Emmanuele DiBenedetto
    Pages 1-15
  3. Emmanuele DiBenedetto
    Pages 17-63
  4. Emmanuele DiBenedetto
    Pages 65-121
  5. Emmanuele DiBenedetto
    Pages 123-170
  6. Emmanuele DiBenedetto
    Pages 171-219
  7. Emmanuele DiBenedetto
    Pages 221-274
  8. Emmanuele DiBenedetto
    Pages 275-324
  9. Emmanuele DiBenedetto
    Pages 375-422
  10. Emmanuele DiBenedetto
    Pages 423-468
  11. Back Matter
    Pages 469-488

About this book

Introduction

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.

Keywords

Maxima Maximum analysis/pdes applications of mathematics bounded mean oscillation calculus differential equation functional analysis linear optimization measure measure theory

Authors and affiliations

  • Emmanuele DiBenedetto
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Bibliographic information

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