Real Analysis / / by Emmanuele DiBenedetto

• Autore: DiBenedetto Emmanuele
• Pubblicazione: New York, NY : , : Springer New York : , : Imprint : Birkhäuser, , 2016
• Edizione: 2nd ed. 2016.
• Descrizione fisica: 1 online resource (XXXII, 596 p. 4 illus.)
• Disciplina: 515.353
• Soggetto topico: Measure theory; Mathematical optimization; Calculus of variations; Differential equations; Approximation theory; Mathematics; Measure and Integration; Calculus of Variations and Optimization; Differential Equations; Approximations and Expansions; Applications of Mathematics
• Nota di contenuto: Intro -- Preface to the Second Edition -- Preface to the First Edition -- Acknowledgments -- Contents -- 1 Preliminaries -- 1 Countable Sets -- 2 The Cantor Set -- 3 Cardinality -- 3.1 Some Examples -- 4 Cardinality of Some Infinite Cartesian Products -- 5 Orderings, the Maximal Principle, and the Axiom of Choice -- 6 Well Ordering -- 6.1 The First Uncountable -- 1c Countable Sets -- 2c The Cantor Set -- 2.1c A Generalized Cantor Set of Positive Measure -- 2.2c A Generalized Cantor Set of Measure Zero -- 2.3c Perfect Sets -- 3c Cardinality -- 2 Topologies and Metric Spaces -- 1 Topological Spaces -- 1.1 Hausdorff and Normal Spaces -- 2 Urysohn's Lemma -- 3 The Tietze Extension Theorem -- 4 Bases, Axioms of Countability and Product Topologies -- 4.1 Product Topologies -- 5 Compact Topological Spaces -- 5.1 Sequentially Compact Topological Spaces -- 6 Compact Subsets of mathbbRN -- 7 Continuous Functions on Countably Compact Spaces -- 8 Products of Compact Spaces -- 9 Vector Spaces -- 9.1 Convex Sets -- 9.2 Linear Maps and Isomorphisms -- 10 Topological Vector Spaces -- 10.1 Boundedness and Continuity -- 11 Linear Functionals -- 12 Finite Dimensional Topological Vector Spaces -- 12.1 Locally Compact Spaces -- 13 Metric Spaces -- 13.1 Separation and Axioms of Countability -- 13.2 Equivalent Metrics -- 13.3 Pseudo Metrics -- 14 Metric Vector Spaces -- 14.1 Maps Between Metric Spaces -- 15 Spaces of Continuous Functions -- 15.1 Spaces of Continuously Differentiable Functions -- 15.2 Spaces of Hölder and Lipschitz Continuous Functions -- 16 On the Structure of a Complete Metric Space -- 16.1 The Uniform Boundedness Principle -- 17 Compact and Totally Bounded Metric Spaces -- 17.1 Pre-Compact Subsets of X -- 1c Topological Spaces -- 1.12c Connected Spaces -- 1.19c Separation Properties of Topological Spaces. ; 4c Bases, Axioms of Countability and Product Topologies -- 4.10c The Box Topology -- 5c Compact Topological Spaces -- 5.8c The Alexandrov One-Point Compactification of {X -- mathcalU} ([3]) -- 7c Continuous Functions on Countably Compact Spaces -- 7.1c Upper-Lower Semi-continuous Functions -- 7.2c Characterizing Lower-Semi Continuous Functions in mathbbRN -- 7.3c On the Weierstrass-Baire Theorem -- 7.4c On the Assumptions of Dini's Theorem -- 9c Vector Spaces -- 9.3c Hamel Bases -- 9.6c On the Dimension of a Vector Space -- 10c Topological Vector Spaces -- 13c Metric Spaces -- 13.10c The Hausdorff Distance of Sets -- 13.11c Countable Products of Metric Spaces -- 14c Metric Vector Spaces -- 15c Spaces of Continuous Functions -- 15.1c Spaces of Hölder and Lipschitz Continuous Functions -- 16c On the Structure of a Complete Metric Space -- 16.3c Completion of a Metric Space -- 16.4c Some Consequences of the Baire Category Theorem -- 17c Compact and Totally Bounded Metric Spaces -- 17.1c An Application of the Lebesgue Number Lemma -- 3 Measuring Sets -- 1 Partitioning Open Subsets of mathbbRN -- 2 Limits of Sets, Characteristic Functions, and σ-Algebras -- 3 Measures -- 3.1 Finite, σ-Finite, and Complete Measures -- 3.2 Some Examples -- 4 Outer Measures and Sequential Coverings -- 4.1 The Lebesgue Outer Measure in mathbbRN -- 4.2 The Lebesgue--Stieltjes Outer Measure [89, 154] -- 5 The Hausdorff Outer Measure in mathbbRN [71] -- 5.1 Metric Outer Measures -- 6 Constructing Measures from Outer Measures [26] -- 7 The Lebesgue--Stieltjes Measure on mathbbR -- 7.1 Borel Measures -- 8 The Hausdorff Measure on mathbbRN -- 9 Extending Measures from Semi-algebras to σ-Algebras -- 9.1 On the Lebesgue--Stieltjes and Hausdorff Measures -- 10 Necessary and Sufficient Conditions for Measurability -- 11 More on Extensions from Semi-algebras to σ-Algebras. ; 12 The Lebesgue Measure of Sets in mathbbRN -- 12.1 A Necessary and Sufficient Condition of Measurability -- 13 Vitali's Nonmeasurable Set [168] -- 14 Borel Sets, Measurable Sets, and Incomplete Measures -- 14.1 A Continuous Increasing Function f:[0,1]to[0,1] -- 14.2 On the Preimage of a Measurable Set -- 14.3 Proof of Propositions 14.1 and 14.2 -- 15 Borel Measures -- 16 Borel, Regular, and Radon Measures -- 16.1 Regular Borel Measures -- 16.2 Radon Measures -- 17 Vitali Coverings -- 18 The Besicovitch Covering Theorem -- 19 Proof of Proposition 18.1 -- 20 The Besicovitch Measure-Theoretical Covering Theorem -- 1c Partitioning Open Subsets of mathbbRN -- 2c Limits of Sets, Characteristic Functions and σ-Algebras -- 3c Measures -- 3.1c Completion of a Measure Space -- 4c Outer Measures -- 5c The Hausdorff Outer Measure in mathbbRN -- 5.1c The Hausdorff Dimension of a Set EsubsetmathbbRN -- 5.2c The Hausdorff Dimension of the Cantor Set is ln 2/ln 3 -- 8c The Hausdorff Measure in mathbbRN -- 8.1c Hausdorff Outer Measure of the Lipschitz Image of a Set -- 8.2c Hausdorff Dimension of Graphs of Lipschitz Functions -- 9c Extending Measures from Semi-algebras to σ-Algebras -- 9.1c Inner and Outer Continuity of λ on Some Algebra mathcalQ -- 10c More on Extensions from Semi-algebras to σ-Algebras -- 10.1c Self-extensions of Measures -- 10.2c Nonunique Extensions of Measures λ on Semi-algebras -- 12c The Lebesgue Measure of Sets in mathbbRN -- 12.1c Inner Measure and Measurability -- 12.2c The Peano--Jordan Measure of Bounded Sets in mathbbRN -- 12.3c Lipschitz Functions and Measurability -- 13c Vitali's Nonmeasurable Set -- 14c Borel Sets, Measurable Sets and Incomplete Measures -- 16c Borel, Regular and Radon Measures -- 16.1c Regular Borel Measures -- 16.2c Regular Outer Measures -- 17c Vitali Coverings. ; 17.1c Pointwise and Measure-Theoretical Vitali Coverings -- 18c The Besicovitch Covering Theorem -- 18.1c The Besicovitch Theorem for Unbounded E -- 18.2c The Besicovitch Measure-Theoretical Inner Covering of Open Sets EsubsetmathbbRN -- 18.3c A Simpler Form of the Besicovitch Theorem -- 18.4c Another Besicovitch-Type Covering -- 4 The Lebesgue Integral -- 1 Measurable Functions -- 2 The Egorov--Severini Theorem [39, 145] -- 2.1 The Egorov--Severini Theorem in mathbbRN -- 3 Approximating Measurable Functions by Simple Functions -- 4 Convergence in Measure (Riesz [125], Fisher [46]) -- 5 Quasicontinuous Functions and Lusin's Theorem -- 6 Integral of Simple Functions ([87]) -- 7 The Lebesgue Integral of Nonnegative Functions -- 8 Fatou's Lemma and the Monotone Convergence Theorem -- 9 More on the Lebesgue Integral -- 10 Convergence Theorems -- 11 Absolute Continuity of the Integral -- 12 Product of Measures -- 13 On the Structure of (mathcalAtimesmathcalB) -- 14 The Theorem of Fubini--Tonelli -- 14.1 The Tonelli Version of the Fubini Theorem -- 15 Some Applications of the Fubini--Tonelli Theorem -- 15.1 Integrals in Terms of Distribution Functions -- 15.2 Convolution Integrals -- 15.3 The Marcinkiewicz Integral ([101, 102]) -- 16 Signed Measures and the Hahn Decomposition -- 17 The Radon-Nikodým Theorem -- 17.1 Sublevel Sets of a Measurable Function -- 17.2 Proof of the Radon-Nikodým Theorem -- 18 Decomposing Measures -- 18.1 The Jordan Decomposition -- 18.2 The Lebesgue Decomposition -- 18.3 A General Version of the Radon-Nikodým Theorem -- 1c Measurable Functions -- 1.1c Sublevel Sets -- 2c The Egorov--Severini Theorem -- 3c Approximating Measurable Functions by Simple Functions -- 4c Convergence in Measure -- 7c The Lebesgue Integral of Nonnegative Measurable Functions -- 7.1c Comparing the Lebesgue Integral with the Peano-Jordan Integral. ; 7.2c On the Definition of the Lebesgue Integral -- 9c More on the Lebesgue Integral -- 10c Convergence Theorems -- 10.1c Another Version of Dominated Convergence -- 11c Absolute Continuity of the Integral -- 12c Product of Measures -- 12.1c Product of a Finite Sequence of Measure Spaces -- 13c On the Structure of (mathcalAtimesmathcalB) -- 13.1c Sections and Their Measure -- 14c The Theorem of Fubini--Tonelli -- 15c Some Applications of the Fubini--Tonelli Theorem -- 15.1c Integral of a Function as the ``Area Under the Graph'' -- 15.2c Distribution Functions -- 17c The Radon-Nikodým Theorem -- 18c A Proof of the Radon-Nikodým Theorem When Both μ and ν Are σ-Finite -- 5 Topics on Measurable Functions of Real Variables -- 1 Functions of Bounded Variation ([78]) -- 2 Dini Derivatives ([37]) -- 3 Differentiating Functions of Bounded Variation -- 4 Differentiating Series of Monotone Functions -- 5 Absolutely Continuous Functions ([91, 169]) -- 6 Density of a Measurable Set -- 7 Derivatives of Integrals -- 8 Differentiating Radon Measures -- 9 Existence and Measurability of Dμν -- 9.1 Proof of Proposition 9.2 -- 10 Representing Dμν -- 10.1 Representing Dμν for νllμ -- 10.2 Representing Dμν for νperpμ -- 11 The Lebesgue-Besicovitch Differentiation Theorem -- 11.1 Points of Density -- 11.2 Lebesgue Points of an Integrable Function -- 12 Regular Families -- 13 Convex Functions -- 14 The Jensen's Inequality -- 15 Extending Continuous Functions -- 15.1 The Concave Modulus of Continuity of f -- 16 The Weierstrass Approximation Theorem -- 17 The Stone-Weierstrass Theorem -- 18 Proof of the Stone-Weierstrass Theorem -- 18.1 Proof of Stone's Theorem -- 19 The Ascoli-Arzelà Theorem -- 19.1 Pre-compact Subsets of C(barE) -- 1c Functions of Bounded Variations -- 1.1c The Function of The Jumps -- 1.2c The Space BV[a,b] -- 2c Dini Derivatives. ; 2.1c A Continuous, Nowhere Differentiable Function ([167]).
• Sommario/riassunto: The second edition of this classic textbook presents a rigorous and self-contained introduction to real analysis with the goal of providing a solid foundation for future coursework and research in applied mathematics. Written in a clear and concise style, it covers all of the necessary subjects as well as those often absent from standard introductory texts. Each chapter features a “Problems and Complements” section that includes additional material that briefly expands on certain topics within the chapter and numerous exercises for practicing the key concepts. The first eight chapters explore all of the basic topics for training in real analysis, beginning with a review of countable sets before moving on to detailed discussions of measure theory, Lebesgue integration, Banach spaces, functional analysis, and weakly differentiable functions. More topical applications are discussed in the remaining chapters, such as maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions. This second edition has been completely revised and updated and contains a variety of new content and expanded coverage of key topics, such as new exercises on the calculus of distributions, a proof of the Riesz convolution, Steiner symmetrization, and embedding theorems for functions in Sobolev spaces. Ideal for either classroom use or self-study, Real Analysis is an excellent textbook both for students discovering real analysis for the first time and for mathematicians and researchers looking for a useful resource for reference or review. Praise for the First Edition: “[This book] will be extremely useful as a text. There is certainly enough material for a year-long graduate course, but judicious selection would make it possible to use this most appealing book in a one-semester course for well-prepared students.” —Mathematical Reviews.
• Titolo autorizzato: Real analysis
• ISBN: 1-4939-4005-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910254087103321
• Lo trovi qui: Univ. Federico II
• Serie: Birkhäuser Advanced Texts Basler Lehrbücher, . 2296-4894