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E-raamat: Functional Analysis

(Technische Universiteit Delft, The Netherlands)
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Functional AnalysisSuurem pilt
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This comprehensive introduction to functional analysis covers both the abstract theory and applications to spectral theory, the theory of partial differential equations, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas, and a mathematical treatment of states and observables in quantum mechanics. The book is accessible to graduate students with basic knowledge of topology, real and complex analysis, and measure theory. With carefully written out proofs, more than 300 problems, and appendices covering the prerequisites, this self-contained volume can be used as a text for various courses at the graduate level and as a reference text for researchers in the field.

This comprehensive introduction to functional analysis covers both the abstract theory and applications. With carefully written out proofs, more than 300 problems, and appendices covering prerequisites, this self-contained volume can be used as a text for a graduate-level course and as a reference text for researchers in the field.

Arvustused

'One of its strengths is that it is a genuine textbook rather than a reference text. It is highly readable and pedagogical, giving a good level of detail in proofs, but staying concise and keeping its story clear rather than being encyclopedic. Another strength of the textbook is that it is well motivated by applications of functional analysis to other areas of mathematics, with a special emphasis on partial differential equations and quantum mechanics throughout the book.' Pierre Portal, zbMATH Open 'Everything is beautifully and clearly expressed. In short, highly recommended!' Klaas Landsman, Nieuw Archief voor Wiskunde

Muu info

A comprehensive, graduate-level introduction to functional analysis covering both the theory and main applications, with over 300 exercises.
Preface ix
Notation and Conventions xi
1 Banach Spaces
1(32)
1.1 Banach Spaces
2(7)
1.2 Bounded Operators
9(9)
1.3 Finite-Dimensional Spaces
18(3)
1.4 Compactness
21(2)
1.5 Integration in Banach Spaces
23(10)
Problems
28(5)
2 The Classical Banach Spaces
33(54)
2.1 Sequence Spaces
33(1)
2.2 Spaces of Continuous Functions
34(13)
2.3 Spaces of Integrable Functions
47(17)
2.4 Spaces of Measures
64(9)
2.5 Banach Lattices
73(14)
Problems
77(10)
3 Hilbert Spaces
87(28)
3.1 Hilbert Spaces
87(5)
3.2 Orthogonal Complements
92(3)
3.3 The Riesz Representation Theorem
95(2)
3.4 Orthonormal Systems
97(3)
3.5 Fxamples
100(15)
Problems
106(9)
4 Duality
115(56)
4.1 Duals of the Classical Banach Spaces
115(13)
4.2 The Hahn-Banach Extension Theorem
128(9)
4.3 Adjoint Operators
137(5)
4.4 The Hahn-Banach Separation Theorem
142(2)
4.5 The Krein-Milman Theorem
144(3)
4.6 The Weak and Weak Topologies
147(5)
4.7 The Banach-Alaoglu Theorem
152(19)
Problems
163(8)
5 Bounded Operators
171(38)
5.1 The Uniform Boundedness Theorem
171(3)
5.2 The Open Mapping Theorem
174(2)
5.3 The Closed Graph Theorem
176(2)
5.4 The Closed Range Theorem
178(2)
5.5 The Fourier Transform
180(10)
5.6 The Hilbert Transform
190(2)
5.7 Interpolation
192(17)
Problems
201(8)
6 Spectral Theory
209(18)
6.1 Spectrum and Resolvent
209(7)
6.2 The holomorphic Functional Calculus
216(11)
Problems
224(3)
7 Compact Operators
227(28)
7.1 Compact Operators
227(3)
7.2 The Riesz-Schauder Theorem
230(4)
7.3 Fredholm Theory
234(21)
Problems
252(3)
8 Bounded Operators on Hilbert Spaces
255(26)
8.1 Selfadjoint, Unitary, and Normal Operators
255(11)
8.2 The Continuous Functional Calculus
266(6)
8.3 The Sz.-Nagy Dilation Theorem
272(9)
Problems
277(4)
9 The Spectral Theorem for Bounded Normal Operators
281(30)
9.1 The Spectral Theorem for Compact Normal Operators
281(4)
9.2 Projection-Valued Measures
285(2)
9.3 The Bounded Functional Calculus
287(6)
9.4 The Spectral Theorem for Bounded Normal Operators
293(8)
9.5 The Von Neumann Bicommutant Theorem
301(4)
9.6 Application to Orthogonal Polynomials
305(6)
Problems
308(3)
10 The Spectral Theorem for Unbounded Normal Operators
311(36)
10.1 Unbounded Operators
311(11)
10.2 Unbounded Selfadjoint Operators
322(4)
10.3 Unbounded Normal Operators
326(6)
10.4 The Spectral Theorem for Unbounded Normal Operators
332(15)
Problems
344(3)
11 Boundary Value Problems
347(52)
11.1 Sobolev Spaces
347(25)
11.2 The Poisson Problem -Δu = f
372(13)
11.3 The Lax-Milgram Theorem
385(14)
Problems
388(11)
12 Forms
399(28)
12.1 Forms
399(11)
12.2 The Friedrichs Extension Theorem
410(1)
12.3 The Dirichlet and Neumann Laplacians
411(4)
12.4 The Poisson Problem Revisited
415(1)
12.5 Weyl's Theorem
416(11)
Problems
425(2)
13 Semigroups of Linear Operators
427(80)
13.1 Co-Semigroups
427(12)
13.2 The Hille-Yosida Theorem
439(8)
13.3 The Abstract Cauchy Problem
447(8)
13.4 Analytic Semigroups
455(18)
13.5 Stone's Theorem
473(2)
13.6 Examples
475(23)
13.7 Semigroups Generated by Normal Operators
498(9)
Problems
500(7)
14 Trace Class Operators
507(36)
14.1 Hilbert-Schmidt Operators
507(3)
14.2 Trace Class Operators
510(11)
14.3 Trace Duality
521(2)
14.4 The Partial Trace
523(3)
14.5 Trace Formulas
526(17)
Problems
539(4)
15 States and Observables
543(78)
15.1 States and Observables in Classical Mechanics
543(2)
15.2 States and Observables in Quantum Mechanics
545(16)
15.3 Positive Operator-Valued Measures
561(13)
15.4 Hidden Variables
574(3)
15.5 Symmetries
577(21)
15.6 Second Quantisation
598(23)
Problems
617(4)
Appendix A Zom's Lemma 621(2)
Appendix B Tensor Products 623(4)
Appendix C Topological Spaces 627(10)
Appendix D Metric Spaces 637(10)
Appendix E Measure Spaces 647(14)
Appendix F Integration 661(10)
Appendix G Notes 671(20)
References 691(10)
Index 701
Jan van Neerven holds an Antoni van Leeuwenhoek professorship at Delft University of Technology. Author of four books and more than 100 peer-reviewed articles, he is a leading expert in functional analysis and operator theory and their applications in stochastic analysis and the theory of partial differential equations.
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