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El. knyga: Functional Analysis, Spectral Theory, and Applications

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  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 276
  • Išleidimo metai: 21-Nov-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319585406
Kitos knygos pagal šią temą:
Functional Analysis, Spectral Theory, and ApplicationsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 276
  • Išleidimo metai: 21-Nov-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319585406
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This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory.

In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao’s approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study.

Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics.

Recenzijos

All chapters end with a very useful list of additional topics and suggestions for further reading. The book also contains an appendix on set theory and topology, another one on measure theory . The book is carefully written and provides an interesting introduction to functional analysis with a wealth of both classical and more recent applications. (Michael M. Neumann, Mathematical Reviews, July, 2018)

This is an attractive new textbook in functional analysis, aimed at graduate students. the large amount of material covered in this book as well its overall readability, makes it useful as a reference as well as a potential graduate textbook. If you like functional analysis, teach it, or use it in your work, this book certainly merits a careful look. (Mark Hunacek, MAA Reviews, January, 2018).

The present book is different from the usual textbooks on functional analysis: it does not only cover the basic material, but also a number of advanced topics which cannot be found in many other books on the subject. The text is suitable for self-study as well as for the preparation of lectures and seminars. this is a highly recommendable book for students and researchers alike who are interested in functional analysis and its broad applications. (Jan-David Hardtke, zbMATH 1387.46001, 2018)

1 Motivation
1(14)
1.1 From Even and Odd Functions to Group Representations
1(4)
1.2 Partial Differential Equations and the Laplace Operator
5(7)
1.2.1 The Heat Equation
7(3)
1.2.2 The Wave Equation
10(1)
1.2.3 The Mantegna Fresco
11(1)
1.3 What is Spectral Theory?
12(1)
1.4 The Prime Number Theorem
13(1)
1.5 Further Topics
14(1)
2 Norms and Banach Spaces
15(56)
2.1 Norms and Semi-Norms
15(11)
2.1.1 Normed Vector Spaces
16(5)
2.1.2 Semi-Norms and Quotient Norms
21(2)
2.1.3 Isometries are Affine
23(3)
2.1.4 A Comment on Notation
26(1)
2.2 Banach Spaces
26(13)
2.2.1 Proofs of Completeness
29(7)
2.2.2 The Completion of a Normed Vector Space
36(2)
2.2.3 Non-Compactness of the Unit Ball
38(1)
2.3 The Space of Continuous Functions
39(16)
2.3.1 The Arzela---Ascoli Theorem
40(2)
2.3.2 The Stone---Weierstrass Theorem
42(6)
2.3.3 Equidistribution of a Sequence
48(3)
2.3.4 Continuous Functions in LP Spaces
51(4)
2.4 Bounded Operators and Functionals
55(7)
2.4.1 The Norm of Continuous Functionals on C0(X)
60(1)
2.4.2 Banach Algebras
61(1)
2.5 Ordinary Differential Equations
62(8)
2.5.1 The Volterra Equation
63(3)
2.5.2 The Sturm---Liouville Equation
66(4)
2.6 Further Topics
70(1)
3 Hilbert Spaces, Fourier Series, Unitary Representations
71(50)
3.1 Hilbert Spaces
71(15)
3.1.1 Definitions and Elementary Properties
71(4)
3.1.2 Convex Sets in Uniformly Convex Spaces
75(8)
3.1.3 An Application to Measure Theory
83(3)
3.2 Orthonormal Bases and Gram---Schmidt
86(5)
3.2.1 The Non-Separable Case
90(1)
3.3 Fourier Series on Compact Abelian Groups
91(4)
3.4 Fourier Series on Td
95(11)
3.4.1 Convolution on the Torus
97(2)
3.4.2 Dirichlet and Fejer Kernels
99(5)
3.4.3 Differentiability and Fourier Series
104(2)
3.5 Group Actions and Representations
106(14)
3.5.1 Group Actions and Unitary Representations
107(3)
3.5.2 Unitary Representations of Compact Abelian Groups
110(1)
3.5.3 The Strong (Riemann) Integral
111(2)
3.5.4 The Weak (Lebesgue) Integral
113(2)
3.5.5 Proof of the Weight Decomposition
115(3)
3.5.6 Convolution
118(2)
3.6 Further Topics
120(1)
4 Uniform Boundedness and the Open Mapping Theorem
121(14)
4.1 Uniform Boundedness
121(5)
4.1.1 Uniform Boundedness and Fourier Series
123(3)
4.2 The Open Mapping and Closed Graph Theorems
126(7)
4.2.1 Baire Category
126(2)
4.2.2 Proof of the Open Mapping Theorem
128(2)
4.2.3 Consequences: Bounded Inverses and Closed Graphs
130(3)
4.3 Further Topics
133(2)
5 Sobolev Spaces and Dirichlet's Boundary Problem
135(32)
5.1 Sobolev Spaces and Embedding on the Torus
135(5)
5.1.1 L2 Sobolev Spaces on Td
135(3)
5.1.2 The Sobolev Embedding Theorem on Td
138(2)
5.2 Sobolev Spaces on Open Sets
140(12)
5.2.1 Examples
144(2)
5.2.2 Restriction Operators and Traces
146(3)
5.2.3 Sobolev Embedding in the Interior
149(3)
5.3 Dirichlet's Boundary Value Problem and Elliptic Regularity
152(13)
5.3.1 The Semi-Inner Product
153(2)
5.3.2 Elliptic Regularity for the Laplace Operator
155(5)
5.3.3 Dirichlet's Boundary Value Problem
160(5)
5.4 Further Topics
165(2)
6 Compact Self-Adjoint Operators, Laplace Eigenfunctions
167(42)
6.1 Compact Operators
168(6)
6.1.1 Integral Operators are Often Compact
170(4)
6.2 Spectral Theory of Self-Adjoint Compact Operators
174(9)
6.2.1 The Adjoint Operator
175(1)
6.2.2 The Spectral Theorem
176(2)
6.2.3 Proof of the Spectral Theorem
178(3)
6.2.4 Variational Characterization of Eigenvalues
181(2)
6.3 Trace-Class Operators
183(13)
6.4 Eigenfunctions for the Laplace Operator
196(12)
6.4.1 Right Inverse and Compactness on the Torus
197(1)
6.4.2 A Self-Adjoint Compact Right Inverse
198(1)
6.4.3 Eigenfunctions on a Drum
199(2)
6.4.4 Weyl's Law
201(7)
6.5 Further Topics
208(1)
7 Dual Spaces
209(44)
7.1 The Hahn--Banach Theorem and its Consequences
209(8)
7.1.1 The Hahn--Banach Lemma and Theorem
209(3)
7.1.2 Consequences of the Hahn--Banach Theorem
212(1)
7.1.3 The Bidual
213(1)
7.1.4 An Application of the Spanning Criterion
214(3)
7.2 Banach Limits, Amenable Groups, Banach-Tarski
217(10)
7.2.1 Banach Limits
217(1)
7.2.2 Amenable Groups
218(5)
7.2.3 The Banach-Tarski Paradox
223(4)
7.3 The Duals of Lpμ (X)
227(12)
7.3.1 The Dual of L1μ(X)
228(2)
7.3.2 The Dual of L4pμ (X) for p > 1
230(3)
7.3.3 Riesz--Thorin Interpolation
233(6)
7.4 Riesz Representation: The Dual of C(X)
239(13)
7.4.1 Uniqueness
240(1)
7.4.2 Totally Disconnected Compact Spaces
240(3)
7.4.3 Compact Spaces
243(3)
7.4.4 Locally Compact σ-Compact Metric Spaces
246(2)
7.4.5 Continuous Linear Functionals on C0(X)
248(4)
7.5 Further Topics
252(1)
8 Locally Convex Vector Spaces
253(60)
8.1 Weak Topologies and the Banach--Alaoglu Theorem
253(8)
8.1.1 Weak* Compactness of the Unit Ball
256(1)
8.1.2 More Properties of the Weak and Weak* Topologies
257(3)
8.1.3 Analytic Functions and the Weak Topology
260(1)
8.2 Applications of Weak* Compactness
261(29)
8.2.1 Equidistribution
262(8)
8.2.2 Elliptic Regularity for the Laplace Operator
270(8)
8.2.3 Elliptic Regularity at the Boundary
278(12)
8.3 Topologies on the space of bounded operators
290(2)
8.4 Locally Convex Vector Spaces
292(4)
8.5 Distributions as Generalized Functions
296(2)
8.6 Convex Sets
298(13)
8.6.1 Extreme Points and the Krein--Milman Theorem
301(3)
8.6.2 Choquet's Theorem
304(7)
8.7 Further Topics
311(2)
9 Unitary Operators and Flows, Fourier Transform
313(40)
9.1 Spectral Theory of Unitary Operators
313(16)
9.1.1 Herglotz's Theorem for Positive-Definite Sequences
314(2)
9.1.2 Cyclic Representations and the Spectral Theorem
316(4)
9.1.3 Spectral Measures
320(3)
9.1.4 Functional Calculus for Unitary Operators
323(3)
9.1.5 An Application of Spectral Theory to Dynamics
326(3)
9.2 The Fourier Transform
329(15)
9.2.1 The Fourier Transform on L1(Rd)
331(6)
9.2.2 The Fourier Transform on L2(Rd)
337(3)
9.2.3 The Fourier Transform, Smoothness, Schwartz Space
340(2)
9.2.4 The Uncertainty Principle
342(2)
9.3 Spectral Theory of Unitary Flows
344(8)
9.3.1 Positive-Definite Functions; Cyclic Representations
344(2)
9.3.2 The Case G = Rd
346(4)
9.3.3 Stone's Theorem
350(2)
9.4 Further Topics
352(1)
10 Locally Compact Groups, Amenability, Property (T)
353(56)
10.1 Haar Measure
353(8)
10.2 Amenable Groups
361(14)
10.2.1 Definitions and Main Theorem
362(1)
10.2.2 Proof of Theorem 10.15
363(8)
10.2.3 A More Uniform Fømer Set
371(2)
10.2.4 Further Equivalences and Properties
373(2)
10.3 Property (T)
375(25)
10.3.1 Definitions and First Properties
375(2)
10.3.2 Main Theorems
377(1)
10.3.3 Proof of Kazdan's Property (T), Connected Case
378(6)
10.3.4 Proof of Kazdan's Property (T), Discrete Case
384(8)
10.3.5 Iwasawa Decomposition and Geometry of Numbers
392(8)
10.4 Highly Connected Networks: Expanders
400(8)
10.4.1 Constructing an Explicit Expander Family
406(2)
10.5 Further Topics
408(1)
11 Banach Algebras and the Spectrum
409(24)
11.1 The Spectrum and Spectral Radius
409(8)
11.1.1 The Geometric Series and its Consequences
411(2)
11.1.2 Using Cauchy Integration
413(4)
11.2 C*-algebras
417(1)
11.3 Commutative Banach Algebras and their Gelfand Duals
418(7)
11.3.1 Commutative Unital Banach Algebras
419(2)
11.3.2 Commutative Banach Algebras without a Unit
421(1)
11.3.3 The Gelfand Transform
422(1)
11.3.4 The Gelfand Transform for Commutative C*-algebras
423(2)
11.4 Locally Compact Abelian Groups
425(6)
11.4.1 The Pontryagin Dual
428(3)
11.5 Further Topics
431(2)
12 Spectral Theory and Functional Calculus
433(54)
12.1 Definitions and Basic Lemmas
433(4)
12.1.1 Decomposing the Spectrum
433(3)
12.1.2 The Numerical Range
436(1)
12.1.3 The Essential Spectrum
437(1)
12.2 The Spectrum of a Tree
437(6)
12.2.1 The Correct Upper Bound for the Summing Operator
439(2)
12.2.2 The Spectrum of S
441(1)
12.2.3 No Eigenvectors on the Tree
442(1)
12.3 Main Goals: The Spectral Theorem and Functional Calculus
443(3)
12.4 Self-Adjoint Operators
446(13)
12.4.1 Continuous Functional Calculus
447(4)
12.4.2 Corollaries to the Continuous Functional Calculus
451(2)
12.4.3 Spectral Measures
453(1)
12.4.4 The Spectral Theorem for Self-Adjoint Operators
454(4)
12.4.5 Consequences for Unitary Representations
458(1)
12.5 Commuting Normal Operators
459(2)
12.6 Spectral Measures and the Measurable Functional Calculus
461(7)
12.6.1 Non-Diagonal Spectral Measures
461(1)
12.6.2 The Measurable Functional Calculus
462(6)
12.7 Projection-Valued Measures
468(5)
12.8 Locally Compact Abelian Groups and Pontryagin Duality
473(12)
12.8.1 The Spectral Theorem for Unitary Representations
474(3)
12.8.2 Characters Separate Points
477(1)
12.8.3 The Plancherel Formula
478(5)
12.8.4 Pontryagin Duality
483(2)
12.9 Further Topics
485(2)
13 Self-Adjoint and Symmetric Operators
487(16)
13.1 Examples and Definitions
487(4)
13.2 Operators of the Form T*T
491(4)
13.3 Self-Adjoint Operators
495(3)
13.4 Symmetric Operators
498(4)
13.4.1 The Friedrichs Extension
499(1)
13.4.2 Cayley Transform and Deficiency Indices
500(2)
13.5 Further Topics
502(1)
14 The Prime Number Theorem
503(34)
14.1 Two Reformulations
503(4)
14.2 The Selberg Symmetry Formula and Banach Algebra Norm
507(16)
14.2.1 Dirichlet Convolution and Mobius Inversion
507(2)
14.2.2 The Selberg Symmetry Formula
509(5)
14.2.3 Measure-Theoretic Reformulation
514(3)
14.2.4 A Density Function and the Continuity Bound
517(1)
14.2.5 Mertens' Theorem
518(2)
14.2.6 Completing the Proof
520(3)
14.3 Non-Trivial Spectrum of the Banach Algebra
523(1)
14.4 Trivial Spectrum of the Banach Algebra
524(2)
14.5 Primes in Arithmetic Progressions
526(11)
14.5.1 Non-Vanishing of Dirichlet L-function at 1
529(8)
Appendix A Set Theory and Topology
537(14)
A.1 Set Theory and the Axiom of Choice
537(1)
A.2 Basic Definitions in Topology
538(3)
A.3 Inducing Topologies
541(4)
A.4 Compact Sets and Tychonoff's Theorem
545(2)
A.5 Normal Spaces
547(4)
Appendix B Measure Theory
551(12)
B.1 Basic Definitions and Measurability
551(3)
B.2 Properties of the Integral
554(2)
B.3 The p-Norm
556(2)
B.4 Near-Continuity of Measurable Functions
558(3)
B.5 Signed Measures
561(2)
Hints for Selected Problems
563(26)
Notes
589(4)
References
593(7)
Notation
598(2)
General Index 600
Manfred Einsiedler studied mathematics at the University of Vienna and has been a Professor at the ETH Zürich since 2009. He was an invited speaker at the 2008 European Mathematical Congress in Amsterdam and the 2010 International Congress of Mathematicians in Hyderabad. His primary research area is ergodic theory with connections to number theory. In cooperation with Lindenstrauss and Katok, Einsiedler made significant progress towards the Littlewood conjecture.   

Thomas Ward studied mathematics at the University of Warwick and is Deputy Vice-Chancellor for student education at the University of Leeds. He works in ergodic theory and number theory, and has written several monographs, including Heights of Polynomials and Entropy in Algebraic Dynamics with Graham Everest and Ergodic Theory: with a view towards Number Theory with Manfred Einsiedler. 
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