04156nam a22004935i 4500 978-3-030-33143-6 DE-He213 20240828123611.0 cr nn 008mamaa 191129s2020 sz | s |||| 0|eng d 9783030331436 978-3-030-33143-6 10.1007/978-3-030-33143-6 doi QA312-312.5 PBKL bicssc MAT034000 bisacsh PBKL thema 515.42 23 Axler, Sheldon. author. aut http://id.loc.gov/vocabulary/relators/aut 10451 Measure, Integration & Real Analysis [electronic resource] / by Sheldon Axler. 1st ed. 2020. Cham : Springer International Publishing : Imprint: Springer, 2020. XVIII, 411 p. 41 illus., 20 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Graduate Texts in Mathematics, 2197-5612 ; 282 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. Open Access 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 isfreely available online. Measure theory. 10452 Measure and Integration. 10453 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783030331429 Printed edition: 9783030331443 Graduate Texts in Mathematics, 2197-5612 ; 282 https://doi.org/10.1007/978-3-030-33143-6 ZDB-2-SMA ZDB-2-SXMS ZDB-2-SOB 444070 444070