A Guide to Advanced Real Analysis
A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and i...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2009
Mathematical Association of America |
| Ausgabe: | 1 |
| Schriftenreihe: | Dolciani Mathematical Expositions |
| Schlagworte: | |
| ISBN: | 0883853434, 9780883853436 |
| Online-Zugang: | Volltext |
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| Abstract | A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form. |
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| AbstractList | This book is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form. A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form. |
| Author | Folland, Gerald B. |
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| Copyright | Copyright 2009 American Mathematical Society 2009 The Mathematical Association of America (Incorporated) |
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| Notes | Includes bibliographical references and index |
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| Snippet | A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for... This book is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as... |
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| SubjectTerms | Functions of real variables Mathematical analysis Mathematics |
| TableOfContents | Preface
--
Prologue: Notation, Terminology, and Set Theory
--
Topology
--
Measure and Integration: General Theory
--
Measure and Integration: Constructions and Special Examples
--
Rudiments of Functional Analysis
--
Function Spaces
--
Topics in Analysis on Euclidean Space
--
Bibliography Front Matter Preface Table of Contents Prologue: CHAPTER 1: Topology CHAPTER 2: Measure and Integration: CHAPTER 3: Measure and Integration: CHAPTER 4: Rudiments of Functional Analysis CHAPTER 5: Function Spaces CHAPTER 6: Topics in Analysis on Euclidean Space Bibliography Index Back Matter Intro -- Preface -- Contents -- Prologue: Notation, Terminology, and Set Theory -- Numbers -- Sets and mappings -- Zorn's lemma -- 1 Topology -- 1.1 Metric spaces -- 1.2 Topological spaces and continuous maps -- 1.3 Neighborhood bases and convergence -- 1.4 Compactness -- 2 Measure and Integration: General Theory -- 2.1 Measures -- 2.2 Integration -- 2.3 Convergence of functions and convergence of integrals -- 2.4 Product measures and the Fubini-Tonelli theorem -- 2.5 Relations between (signed and complex) measures -- 3 Measure and Integration: Constructions and Special Examples -- 3.1 Construction of measures -- 3.2 Lebesgue measure -- 3.3 Regular Borel measures and functions on the real line -- 3.4 Hausdorff measure -- 3.5 Regular Borel measures on LCH spaces -- 4 Rudiments of Functional Analysis -- 4.1 Normed vector spaces and bounded linear maps -- 4.2 Hilbert spaces -- 4.3 Other topological vector spaces -- 5 Function Spaces -- 5.1 L^p spaces -- 5.2 Spaces of continuous functions -- 6 Topics in Analysis on Euclidean Space -- 6.1 Convolutions -- 6.2 Fourier series and transforms -- 6.3 Distributions -- Bibliography -- Index -- About the Author |
| Title | A Guide to Advanced Real Analysis |
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| Volume | 37 |
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