Kooperativer Bibliotheksverbund

UID:

Format: 1 Online-Ressource (xviii, 411 Seiten) , Illustrationen

ISBN: 9783030331436

Series Statement: Graduate Texts in Mathematics 282

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33142-9

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33144-3

Language: English

Subjects: Economics , Mathematics

Keywords: Mathematik ; Analysis ; Integralrechnung

DOI: 10.1007/978-3-030-33143-6

URL: Volltext  (kostenfrei)

URL: Volltext  (kostenfrei)

URL: Buchcover

Author information: Axler, Sheldon Jay 1949-

UID:

Format: xviii, 411 Seiten : , Illustrationen, Diagramme (teilweise farbig).

ISBN: 978-3-030-33142-9

Series Statement: Graduate texts in mathematics 282

Additional Edition: Erscheint auch als Online-Ausgabe ISBN 978-3-030-33143-6

Language: English

Subjects: Economics , Mathematics

Keywords: Mathematik ; Analysis ; Integralrechnung

URL: Review

Author information: Axler, Sheldon Jay, 1949-,

UID:

Format: 1 Online-Ressource (411 p.)

ISBN: 9783030331436

Series Statement: Graduate Texts in Mathematics

Content: This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online

Note: English

Language: English

URL: kostenfrei

UID:

Format: 1 online resource (411)

Edition: 1st ed. 2020.

ISBN: 3-030-33143-1

Series Statement: Graduate Texts in Mathematics, 282

Content: This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

Note: About the Author -- Preface for Students -- Preface for Instructors -- Acknowledgments -- 1. Riemann Integration -- 2. Measures -- 3. Integration -- 4. Differentiation -- 5. Product Measures -- 6. Banach Spaces -- 7. L^p Spaces -- 8. Hilbert Spaces -- 9. Real and Complex Measures -- 10. Linear Maps on Hilbert Spaces -- 11. Fourier Analysis -- 12. Probability Measures -- Photo Credits -- Bibliography -- Notation Index -- Index -- Colophon: Notes on Typesetting. , English.

Additional Edition: ISBN 3-030-33142-3

Language: English

Subjects: Mathematics

Keywords: Lehrbuch

DOI: 10.1007/978-3-030-33143-6

URL: https://doi.org/10.1007/978-3-030-33143-6

URL: kostenfrei

URL: Cover

UID:

Format: 1 online resource (430 pages)

ISBN: 9783030331436

Series Statement: Graduate Texts in Mathematics Ser. ; v.282

Note: Intro -- About the Author -- Contents -- Preface for Students -- Preface for Instructors -- Acknowledgments -- Riemann Integration -- Review: Riemann Integral -- Exercises 1A -- Riemann Integral Is Not Good Enough -- Exercises 1B -- Measures -- Outer Measure on R -- Motivation and Definition of Outer Measure -- Good Properties of Outer Measure -- Outer Measure of Closed Bounded Interval -- Outer Measure is Not Additive -- Exercises 2A -- Measurable Spaces and Functions -- -Algebras -- Borel Subsets of R -- Inverse Images -- Measurable Functions -- Exercises 2B -- Measures and Their Properties -- Definition and Examples of Measures -- Properties of Measures -- Exercises 2C -- Lebesgue Measure -- Additivity of Outer Measure on Borel Sets -- Lebesgue Measurable Sets -- Cantor Set and Cantor Function -- Exercises 2D -- Convergence of Measurable Functions -- Pointwise and Uniform Convergence -- Egorov's Theorem -- Approximation by Simple Functions -- Luzin's Theorem -- Lebesgue Measurable Functions -- Exercises 2E -- Integration -- Integration with Respect to a Measure -- Integration of Nonnegative Functions -- Monotone Convergence Theorem -- Integration of Real-Valued Functions -- Exercises 3A -- Limits of Integrals & -- Integrals of Limits -- Bounded Convergence Theorem -- Sets of Measure 0 in Integration Theorems -- Dominated Convergence Theorem -- Riemann Integrals and Lebesgue Integrals -- Approximation by Nice Functions -- Exercises 3B -- Differentiation -- Hardy-Littlewood Maximal Function -- Markov's Inequality -- Vitali Covering Lemma -- Hardy-Littlewood Maximal Inequality -- Exercises 4A -- Derivatives of Integrals -- Lebesgue Differentiation Theorem -- Derivatives -- Density -- Exercises 4B -- Product Measures -- Products of Measure Spaces -- Products of -Algebras -- Monotone Class Theorem -- Products of Measures. , Exercises 5A -- Iterated Integrals -- Tonelli's Theorem -- Fubini's Theorem -- Area Under Graph -- Exercises 5B -- Lebesgue Integration on Rn -- Borel Subsets of Rn -- Lebesgue Measure on Rn -- Volume of Unit Ball in Rn -- Equality of Mixed Partial Derivatives Via Fubini's Theorem -- Exercises 5C -- Banach Spaces -- Metric Spaces -- Open Sets, Closed Sets, and Continuity -- Cauchy Sequences and Completeness -- Exercises 6A -- Vector Spaces -- Integration of Complex-Valued Functions -- Vector Spaces and Subspaces -- Exercises 6B -- Normed Vector Spaces -- Norms and Complete Norms -- Bounded Linear Maps -- Exercises 6C -- Linear Functionals -- Bounded Linear Functionals -- Discontinuous Linear Functionals -- Hahn-Banach Theorem -- Exercises 6D -- Consequences of Baire's Theorem -- Baire's Theorem -- Open Mapping Theorem and Inverse Mapping Theorem -- Closed Graph Theorem -- Principle of Uniform Boundedness -- Exercises 6E -- Lp Spaces -- Lp() -- Hölder's Inequality -- Minkowski's Inequality -- Exercises 7A -- Lp() -- Definition of Lp() -- Lp() Is a Banach Space -- Duality -- Exercises 7B -- Hilbert Spaces -- Inner Product Spaces -- Inner Products -- Cauchy-Schwarz Inequality and Triangle Inequality -- Exercises 8A -- Orthogonality -- Orthogonal Projections -- Orthogonal Complements -- Riesz Representation Theorem -- Exercises 8B -- Orthonormal Bases -- Bessel's Inequality -- Parseval's Identity -- Gram-Schmidt Process and Existence of Orthonormal Bases -- Riesz Representation Theorem, Revisited -- Exercises 8C -- Real and Complex Measures -- Total Variation -- Properties of Real and Complex Measures -- Total Variation Measure -- The Banach Space of Measures -- Exercises 9A -- Decomposition Theorems -- Hahn Decomposition Theorem -- Jordan Decomposition Theorem -- Lebesgue Decomposition Theorem -- Radon-Nikodym Theorem -- Dual Space of Lp(). , Exercises 9B -- Linear Maps on Hilbert Spaces -- Adjoints and Invertibility -- Adjoints of Linear Maps on Hilbert Spaces -- Null Spaces and Ranges in Terms of Adjoints -- Invertibility of Operators -- Exercises 10A -- Spectrum -- Spectrum of an Operator -- Self-adjoint Operators -- Normal Operators -- Isometries and Unitary Operators -- Exercises 10B -- Compact Operators -- The Ideal of Compact Operators -- Spectrum of Compact Operator and Fredholm Alternative -- Exercises 10C -- Spectral Theorem for Compact Operators -- Orthonormal Bases Consisting of Eigenvectors -- Singular Value Decomposition -- Exercises 10D -- Fourier Analysis -- Fourier Series and Poisson Integral -- Fourier Coefficients and Riemann-Lebesgue Lemma -- Poisson Kernel -- Solution to Dirichlet Problem on Disk -- Fourier Series of Smooth Functions -- Exercises 11A -- Fourier Series and Lp of Unit Circle -- Orthonormal Basis for L2 of Unit Circle -- Convolution on Unit Circle -- Exercises 11B -- Fourier Transform -- Fourier Transform on L1(R) -- Convolution on R -- Poisson Kernel on Upper Half-Plane -- Fourier Inversion Formula -- Extending Fourier Transform to L2(R) -- Exercises 11C -- Probability Measures -- Probability Spaces -- Independent Events and Independent Random Variables -- Variance and Standard Deviation -- Conditional Probability and Bayes' Theorem -- Distribution and Density Functions of Random Variables -- Weak Law of Large Numbers -- Exercises 12 -- Photo Credits -- Bibliography -- Notation Index -- Index -- Colophon: Notes on Typesetting.

Additional Edition: Print version: Axler, Sheldon Measure, Integration and Real Analysis Cham : Springer International Publishing AG,c2019 ISBN 9783030331429

Language: English

Keywords: Electronic books.

URL: Click to View

URL: lizenzpflichtig

UID:

Format: 1 online resource

ISBN: 9783030331436 , 3030331431

Series Statement: Graduate Texts in Mathematics Ser. ; v. 282

Content: This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn-Banach Theorem, Hölder's Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

Note: About the Author -- Preface for Students -- Preface for Instructors -- Acknowledgments -- 1. Riemann Integration -- 2. Measures -- 3. Integration -- 4. Differentiation -- 5. Product Measures -- 6. Banach Spaces -- 7. L^p Spaces -- 8. Hilbert Spaces -- 9. Real and Complex Measures -- 10. Linear Maps on Hilbert Spaces -- 11. Fourier Analysis -- 12. Probability Measures -- Photo Credits -- Bibliography -- Notation Index -- Index -- Colophon: Notes on Typesetting.

Additional Edition: Print version: AXLER, SHELDON. MEASURE, INTEGRATION & REAL ANALYSIS. [Place of publication not identified] : SPRINGER NATURE, 2019 ISBN 3030331423

Additional Edition: ISBN 9783030331429

Language: English

URL: ProQuest Ebook Central

URL: SpringerLink

UID:

Format: 1 online resource (411)

Edition: 1st ed. 2020.

ISBN: 3-030-33143-1

Series Statement: Graduate Texts in Mathematics, 282

Content: This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

Note: About the Author -- Preface for Students -- Preface for Instructors -- Acknowledgments -- 1. Riemann Integration -- 2. Measures -- 3. Integration -- 4. Differentiation -- 5. Product Measures -- 6. Banach Spaces -- 7. L^p Spaces -- 8. Hilbert Spaces -- 9. Real and Complex Measures -- 10. Linear Maps on Hilbert Spaces -- 11. Fourier Analysis -- 12. Probability Measures -- Photo Credits -- Bibliography -- Notation Index -- Index -- Colophon: Notes on Typesetting. , English.

Additional Edition: ISBN 3-030-33142-3

Language: English

DOI: 10.1007/978-3-030-33143-6

UID:

Format: 1 online resource (411)

Edition: 1st ed. 2020.

ISBN: 3-030-33143-1

Series Statement: Graduate Texts in Mathematics, 282

Content: This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

Note: About the Author -- Preface for Students -- Preface for Instructors -- Acknowledgments -- 1. Riemann Integration -- 2. Measures -- 3. Integration -- 4. Differentiation -- 5. Product Measures -- 6. Banach Spaces -- 7. L^p Spaces -- 8. Hilbert Spaces -- 9. Real and Complex Measures -- 10. Linear Maps on Hilbert Spaces -- 11. Fourier Analysis -- 12. Probability Measures -- Photo Credits -- Bibliography -- Notation Index -- Index -- Colophon: Notes on Typesetting. , English.

Additional Edition: ISBN 3-030-33142-3

Language: English

DOI: 10.1007/978-3-030-33143-6

UID:

Format: 1 Online-Ressource (xviii, 411 Seiten) : , Illustrationen.

ISBN: 978-3-030-33143-6

Series Statement: Graduate Texts in Mathematics 282

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33142-9

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33144-3

Language: English

Subjects: Economics , Mathematics

Keywords: Mathematik ; Analysis ; Integralrechnung

DOI: 10.1007/978-3-030-33143-6

URL: Volltext  (kostenfrei)

URL: Volltext  (kostenfrei)

URL: http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031661432&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA

Author information: Axler, Sheldon Jay 1949-

UID:

Format: 1 Online-Ressource (xviii, 411 Seiten) : , Illustrationen.

ISBN: 978-3-030-33143-6

Series Statement: Graduate Texts in Mathematics 282

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33142-9

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-33144-3

Language: English

Subjects: Economics , Mathematics

Keywords: Mathematik ; Analysis ; Integralrechnung

DOI: 10.1007/978-3-030-33143-6

URL: Volltext  (kostenfrei)

URL: Volltext  (kostenfrei)

URL: http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031661432&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA

Author information: Axler, Sheldon Jay 1949-