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BL
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| Record Number
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862773
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| Main Entry
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Moroșanu, Gheorghe.
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| Title & Author
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Functional analysis for the applied sciences /\ Gheorghe Moroşanu.
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| Publication Statement
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Cham :: Springer,, ©2019.
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| Series Statement
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Universitext
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| Page. NO
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1 online resource (439 pages)
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| ISBN
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3030271536
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: 9783030271534
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3030271528
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9783030271527
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| Notes
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12.1 Answers to Exercises for Chap. 1
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| Bibliographies/Indexes
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Includes bibliographical references.
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| Contents
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Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Sets -- 1.2 Sequences -- 1.3 Real Numbers -- 1.4 Complex Numbers -- 1.5 Linear Spaces -- 1.6 Exercises -- 2 Metric Spaces -- 2.1 Definitions -- 2.2 Completeness -- 2.3 Compact Sets -- 2.4 Continuous Functions on Compact Sets -- 2.5 The Banach Contraction Principle -- 2.6 Exercises -- 3 The Lebesgue Integral and Lp Spaces -- 3.1 Measurable Sets in Rk -- 3.2 Measurable Functions -- 3.3 The Lebesgue Integral -- 3.4 Lp Spaces -- 3.5 Exercises -- 4 Continuous Linear Operators and Functionals -- 4.1 Definitions, Examples, Operator Norm
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4.2 Main Principles of Functional Analysis -- 4.3 Compact Linear Operators -- 4.4 Linear Functionals, Dual Spaces, Weak Topologies -- 4.5 Exercises -- 5 Distributions, Sobolev Spaces -- 5.1 Test Functions -- 5.2 Friedrichs' Mollification -- 5.3 Scalar Distributions -- 5.3.1 Some Operations with Distributions -- 5.3.2 Convergence in Distributions -- 5.3.3 Differentiation of Distributions -- 5.3.4 Differential Equations for Distributions -- 5.4 Sobolev Spaces -- 5.5 Bochner's Integral -- 5.6 Vector Distributions, Wmp(a, b -- X) Spaces -- 5.7 Exercises -- 6 Hilbert Spaces -- 6.1 Examples
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6.2 Jordan-von Neumann Characterization Theorem -- 6.3 Projections in Hilbert Spaces -- 6.4 The Riesz Representation Theorem -- 6.5 Lax-Milgram Theorem -- 6.6 Fourier Series Expansions -- 6.7 Exercises -- 7 Adjoint, Symmetric, and Self-adjoint LinearOperators -- 7.1 The Adjoint of a Linear Operator -- 7.2 Adjoints of Operators on Hilbert Spaces -- 7.2.1 The Case of Compact Operators -- 7.3 Symmetric Operators and Self-adjoint Operators -- 7.4 Exercises -- 8 Eigenvalues and Eigenvectors -- 8.1 Definition and Examples -- 8.2 Main Results
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8.3 Eigenvalues of -Delta Under the Dirichlet BoundaryCondition -- 8.4 Eigenvalues of -Delta Under the Robin BoundaryCondition -- 8.5 Eigenvalues of -Delta Under the Neumann BoundaryCondition -- 8.6 Some Comments -- 8.7 Exercises -- 9 Semigroups of Linear Operators -- 9.1 Definitions -- 9.2 Some Properties of C0-Semigroups -- 9.3 Uniformly Continuous Semigroups -- 9.4 Groups of Linear Operators. Definitions and Linkto Operator Semigroups -- 9.5 Translation Semigroups -- 9.6 The Hille-Yosida Generation Theorem -- 9.7 The Lumer-Phillips Theorem -- 9.8 The Feller-Miyadera-Phillips Theorem
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9.9 A Perturbation Result -- 9.10 Approximation of Semigroups -- 9.11 The Inhomogeneous Cauchy Problem -- 9.12 Applications -- 9.12.1 The Heat Equation -- 9.12.2 The Wave Equation -- 9.12.3 The Transport Equation -- 9.12.4 The Telegraph System -- 9.13 Exercises -- 10 Solving Linear Evolution Equationsby the Fourier Method -- 10.1 First Order Linear EvolutionEquations -- 10.2 Second Order Linear EvolutionEquations -- 10.3 Examples -- 10.4 Exercises -- 11 Integral Equations -- 11.1 Volterra Equations -- 11.2 Fredholm Equations -- 11.3 Exercises -- 12 Answers to Exercises
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| Abstract
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This advanced graduate textbook presents main results and techniques in Functional Analysis and uses them to explore other areas of mathematics and applications. Special attention is paid to creating appropriate frameworks towards solving significant problems involving differential and integral equations. Exercises at the end of each chapter help the reader to understand the richness of ideas and methods offered by Functional Analysis. Some of the exercises supplement theoretical material, while others relate to the real world. This textbook, with its friendly exposition, focuses on different problems in physics and other applied sciences and uniquely provides solutions to most of the exercises. The text is aimed toward graduate students and researchers in applied mathematics, physics, and neighboring fields of science.
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| Subject
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Functional analysis.
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| Subject
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Functional analysis.
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| Dewey Classification
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515/.7
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| LC Classification
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QA320
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