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1. Linear Spaces1

2. Linear Maps8

2.1 Algebra of linear maps8

2.2. Index of a linear map12

3. The Hahn-Banach Theorem19

3.1 The extension theorem19

3.2 Geometric Hahn-Banach theorem21

3.3 Extensions of the Hahn-Banach theorem24

4. Applications of the Hahn-Banach theorem29

4.1 Extension of positive linear functionals29

4.2 Banach limits31

4.3 Finitely additive invariant set functions33

Historical note34

5. Normed Linear Spaces36

5.1 Norms36

5.2 Noncompactness of the unit ball43

5.3 Isometries47

6. Hilbert Space52

6.1 Scalar product52

6.2 Closest point in a closed convex subset54

6.3 Linear functionals56

6.4 Linear span58

7. Applications of Hilbert Space Results63

7.1 Radon-Nikodym theorem63

7.2 Dirichlet's problem65

8. Duals of Normed Linear Spaces72

8.1 Bounded linear functionals72

8.2 Extension of bounded linear functionals74

8.3 Reflexive spaces78

8.4 Support function of a set83

9. Applications of Duality87

9.1 Completeness of weighted powers87

9.2 The Muntz approximation theorem88

9.3 Runge's theorem91

9.4 Dual variational problems in function theory91

9.5 Existence of Green's function94

10. Weak Convergence99

10.1 Uniform boundedness of weakly convergent sequences101

10.2 Weak sequential compactness104

10.3 Weak convergence105

11. Applications of Weak Convergence108

11.1 Approximation of the?? function by continuous functions108

11.2 Divergence of Fourier series109

11.3 Approximate quadrature110

11.4 Weak and strong analyticity of vector-valued functions111

11.5 Existence of solutions of partial differential equations112

11.6 The representation of analytic functions with positive real part115

12. The Weak and Weak Topologies118

13. Locally Convex Topologies and the Krein-Milman Theorem122

13.1 Separation of points by linear functional123

13.2 The Krein-Milman theorem124

13.3 The Stone-Weierstrass theorem126

13.4 Choquet's theorem128

14. Examples of Convex Sets and Their Extreme Points133

14.1 Positive functionals133

14.2 Convex functions135

14.3 Completely monotone functions137

14.4 Theorems of Caratheodory and Bochner141

14.5 A theorem of Krein147

14.6 Positive harmonic functions148

14.7 The Hamburger moment problem150

14.8 G. Birkhoff's conjecture151

14.9 De Finetti's theorem156

14.10 Measure-preserving mappings157

Historical note159

15. Bounded Linear Maps160

15.1 Boundedness and continuity160

15.2 Strong and weak topologies165

15.3 Principle of uniform boundedness166

15.4 Composition of bounded maps167

15.5 The open mapping principle168

Historical note172

16. Examples of Bounded Linear Maps173

16.1 Boundedness of integral operators173

16.2 The convexity theorem of Marcel Riesz177

16.3 Examples of bounded integral operators180

16.4 Solution operators for hyperbolic equations186

16.5 Solution operator for the heat equation188

16.6 Singular integral operators pseudodifferential operators and Fourier integral operators190

17. Banach Algebras and their Elementary Spectral Theory192

17.1 Normed algebras192

17.2 Functional calculus197

18. Gelfand's Theory of Commutative Banach Algebras202

19. Applications of Gelfand's Theory of Commutative Banach Algebras210

19.1 The algebra C(S)210

19.2 Gelfand compactification210

19.3 Absolutely convergent Fourier series212

19.4 Analytic functions in the closed unit disk213

19.5 Analytic functions in the open unit disk214

19.6 Wiener's Tauberian theorem215

19.7 Commutative B-algebras221

Historical note224

20. Examples of Operators and Their Spectra226

20.1 Invertible maps226

20.2 Shifts229

20.3 Volterra integral operators230

20.4 The Fourier transform231

21. Compact Maps233

21.1 Basic properties of compact maps233

21.2 The spectral theory of compact maps238

Historical note244

22. Examples of Compact Operators245

22.1 Compactness criteria245

22.2 Integral operators246

22.3 The inverse of elliptic partial differential operators249

22.4 Operators defined by parabolic equations250

22.5 Almost orthogonal bases251

23. Positive compact operators253

23.1 The spectrum of compact positive operators253

23.2 Stochastic integral operators256

23.3 Inverse of a second order elliptic operator258

24. Fredholm's Theory of Integral Equations260

24.1 The Fredholm determinant and the Fredholm resolvent260

24.2 The multiplicative property of the Fredholm determinant268

24.3 The Gelfand-Levitan-Marchenko equation and Dyson's formula271

25. Invariant Subspaces275

25.1 Invariant subspaces of compact maps275

25.2 Nested invariant subspaces277

26. Harmonic Analysis on a Halfline284

26.1 The Phragmen-Lindelof principle for harmonic functions284

26.2 An abstract Pragmen-Lindelof principle285

26.3 Asymptotic expansion297

27. Index Theory300

27.1 The Noether index301

Historical note305

27.2 Toeplitz operators305

27.3 Hankel operators312

28. Compact Symmetric Operators in Hilbert Space315

29. Examples of Compact Symmetric Operators323

29.1 Convolution323

29.2 The inverse of a differential operator326

29.3 The inverse of partial differential operators327

30. Trace Class and Trace Formula329

30.1 Polar decomposition and singular values329

30.2 Trace class,trace norm,and trace330

30.3 The trace formula334

30.4 The determinant341

30.5 Examples and counterexamples of trace class operators342

30.6 The Poisson summation formula348

30.7 How to express the index of an operator as a difference of traces349

30.8 The Hilbert-Schmidt class352

30.9 Determinant and trace for operator in Banach spaces353

31. Spectral Theory of Symmetric,Normal,and Unitary Operators354

31.1 The spectrum of symmetric operators356

31.2 Functional calculus for symmetric operators358

31.3 Spectral resolution of symmetric operators361

31.4 Absolutely continuous,singular,and point spectra364

31.5 The spectral representation of symmetric operators364

31.6 Spectral resolution of normal operators370

31.7 Spectral resolution of unitary operators372

Historical note375

32. Spectral Theory of Self-Adjoint Operators377

32.1 Spectral resolution378

32.2 Spectral resolution using the Cayley transform389

32.3 A functional calculus for self-adjoint operators390

33. Examples of Self-Adjoint Operators394

33.1 The extension of unbounded symmetric operators394

33.2 Examples of the extension of symmetric operators; deficiency indices397

33.3 The Friedrichs extension402

33.4 The Rellich perturbation theorem406

33.5 The moment problem410

Historical note414

34. Semigroups of Operators416

34.1 Strongly continuous one-parameter semigroups418

34.2 The generation of semigroups424

34.3 The approximation of semigroups427

34.4 Perturbation of semigroups432

34.5 The spectral theory of semigroups434

35. Groups of Unitary Operators440

35.1 Stone's theorem440

35.2 Ergodic theory443

35.3 The Koopman group445

35.4 The wave equation447

35.5 Translation representation448

35.6 The Heisenberg commutation relation455

Historical note459

36. Examples of Strongly Continuous Semigroups461

36.1 Semigroups denned by parabolic equations461

36.2 Semigroups defined by elliptic equations462

36.3 Exponential decay of semigroups465

36.4 The Lax-Phillips semigroup470

36.5 The wave equation in the exterior of an obstacle472

37. Scattering Theory477

37.1 Perturbation theory477

37.2 The wave operators480

37.3 Existence of the wave operators482

37.4 The invariance of wave operators490

37.5 Potential scattering490

37.6 The scattering operator491

Historical note492

37.7 The Lax-Phillips scattering theory493

37.8 The zeros of the scattering matrix499

37.9 The automorphic wave equation500

38. A Theorem of Beurling513

38.1 The Hardy space513

38.2 Beurling's theorem515

38.3 The Titchmarsh convolution theorem523

Historical note525

Texts527

A. Riesz-Kakutani representation theorem529

A.l Positive linear functionals529

A.2 Volume532

A.3 L as a space of functions535

A.4 Measurable sets and measure538

A.5 The Lebesgue measure and integral541

B. Theory of distributions543

B.l Definitions and examples543

B.2 Operations on distributions545

B.3 Local properties of distributions547

B.4 Applications to partial differential equations554

B.5 The Fourier transform558

B.6 Applications of the Fourier transform568

B.7 Fourier series569

C. Zorn's Lemma571

Author Index573

Subject Index577