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20231204023214.0 |
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231204s2019 xx o ||||0 eng d |
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|a 9783030331436
|q (electronic bk.)
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|z 9783030331429
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|a (MiAaPQ)EBC6111862
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|a (Au-PeEL)EBL6111862
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|a (OCoLC)1143638020
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|a MiAaPQ
|b eng
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|e pn
|c MiAaPQ
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|a QA312-312.5
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|a Axler, Sheldon.
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|a Measure, Integration and Real Analysis.
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|a 1st ed.
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| 264 |
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|a Cham :
|b Springer International Publishing AG,
|c 2019.
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|c ©2020.
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|a 1 online resource (430 pages)
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| 336 |
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|a text
|b txt
|2 rdacontent
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|a computer
|b c
|2 rdamedia
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|a online resource
|b cr
|2 rdacarrier
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|a Graduate Texts in Mathematics Series ;
|v v.282
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|a 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.
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|a 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().
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|a 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.
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|a Description based on publisher supplied metadata and other sources.
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|a Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2023. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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| 655 |
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|a Electronic books.
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| 776 |
0 |
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|i Print version:
|a Axler, Sheldon
|t Measure, Integration and Real Analysis
|d Cham : Springer International Publishing AG,c2019
|z 9783030331429
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| 797 |
2 |
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|a ProQuest (Firm)
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| 830 |
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|a Graduate Texts in Mathematics Series
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| 856 |
4 |
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|u https://ebookcentral.proquest.com/lib/matrademy/detail.action?docID=6111862
|z Click to View
|
