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1. Linear Spaces 1

2. Linear Maps 8

2.1 Algebra of linear maps 8

2.2. Index of a linear map 12

3. The Hahn-Banach Theorem 19

3.1 The extension theorem 19

3.2 Geometric Hahn-Banach theorem 21

3.3 Extensions of the Hahn-Banach theorem 24

4. Applications of the Hahn-Banach theorem 29

4.1 Extension of positive linear functionals 29

4.2 Banach limits 31

4.3 Finitely additive invariant set functions 33

Historical note 34

5. Normed Linear Spaces 36

5.1 Norms 36

5.2 Noncompactness of the unit ball 43

5.3 Isometries 47

6. Hilbert Space 52

6.1 Scalar product 52

6.2 Closest point in a closed convex subset 54

6.3 Linear functionals 56

6.4 Linear span 58

7. Applications of Hilbert Space Results 63

7.1 Radon-Nikodym theorem 63

7.2 Dirichlet's problem 65

8. Duals of Normed Linear Spaces 72

8.1 Bounded linear functionals 72

8.2 Extension of bounded linear functionals 74

8.3 Reflexive spaces 78

8.4 Support function of a set 83

9. Applications of Duality 87

9.1 Completeness of weighted powers 87

9.2 The Muntz approximation theorem 88

9.3 Runge's theorem 91

9.4 Dual variational problems in function theory 91

9.5 Existence of Green's function 94

10. Weak Convergence 99

10.1 Uniform boundedness of weakly convergent sequences 101

10.2 Weak sequential compactness 104

10.3 Weak convergence 105

11. Applications of Weak Convergence 108

11.1 Approximation of the?? function by continuous functions 108

11.2 Divergence of Fourier series 109

11.3 Approximate quadrature 110

11.4 Weak and strong analyticity of vector-valued functions 111

11.5 Existence of solutions of partial differential equations 112

11.6 The representation of analytic functions with positive real part 115

12. The Weak and Weak Topologies 118

13. Locally Convex Topologies and the Krein-Milman Theorem 122

13.1 Separation of points by linear functional 123

13.2 The Krein-Milman theorem 124

13.3 The Stone-Weierstrass theorem 126

13.4 Choquet's theorem 128

14. Examples of Convex Sets and Their Extreme Points 133

14.1 Positive functionals 133

14.2 Convex functions 135

14.3 Completely monotone functions 137

14.4 Theorems of Caratheodory and Bochner 141

14.5 A theorem of Krein 147

14.6 Positive harmonic functions 148

14.7 The Hamburger moment problem 150

14.8 G. Birkhoff's conjecture 151

14.9 De Finetti's theorem 156

14.10 Measure-preserving mappings 157

Historical note 159

15. Bounded Linear Maps 160

15.1 Boundedness and continuity 160

15.2 Strong and weak topologies 165

15.3 Principle of uniform boundedness 166

15.4 Composition of bounded maps 167

15.5 The open mapping principle 168

Historical note 172

16. Examples of Bounded Linear Maps 173

16.1 Boundedness of integral operators 173

16.2 The convexity theorem of Marcel Riesz 177

16.3 Examples of bounded integral operators 180

16.4 Solution operators for hyperbolic equations 186

16.5 Solution operator for the heat equation 188

16.6 Singular integral operators pseudodifferential operators and Fourier integral operators 190

17. Banach Algebras and their Elementary Spectral Theory 192

17.1 Normed algebras 192

17.2 Functional calculus 197

18. Gelfand's Theory of Commutative Banach Algebras 202

19. Applications of Gelfand's Theory of Commutative Banach Algebras 210

19.1 The algebra C(S) 210

19.2 Gelfand compactification 210

19.3 Absolutely convergent Fourier series 212

19.4 Analytic functions in the closed unit disk 213

19.5 Analytic functions in the open unit disk 214

19.6 Wiener's Tauberian theorem 215

19.7 Commutative B-algebras 221

Historical note 224

20. Examples of Operators and Their Spectra 226

20.1 Invertible maps 226

20.2 Shifts 229

20.3 Volterra integral operators 230

20.4 The Fourier transform 231

21. Compact Maps 233

21.1 Basic properties of compact maps 233

21.2 The spectral theory of compact maps 238

Historical note 244

22. Examples of Compact Operators 245

22.1 Compactness criteria 245

22.2 Integral operators 246

22.3 The inverse of elliptic partial differential operators 249

22.4 Operators defined by parabolic equations 250

22.5 Almost orthogonal bases 251

23. Positive compact operators 253

23.1 The spectrum of compact positive operators 253

23.2 Stochastic integral operators 256

23.3 Inverse of a second order elliptic operator 258

24. Fredholm's Theory of Integral Equations 260

24.1 The Fredholm determinant and the Fredholm resolvent 260

24.2 The multiplicative property of the Fredholm determinant 268

24.3 The Gelfand-Levitan-Marchenko equation and Dyson's formula 271

25. Invariant Subspaces 275

25.1 Invariant subspaces of compact maps 275

25.2 Nested invariant subspaces 277

26. Harmonic Analysis on a Halfline 284

26.1 The Phragmen-Lindelof principle for harmonic functions 284

26.2 An abstract Pragmen-Lindelof principle 285

26.3 Asymptotic expansion 297

27. Index Theory 300

27.1 The Noether index 301

Historical note 305

27.2 Toeplitz operators 305

27.3 Hankel operators 312

28. Compact Symmetric Operators in Hilbert Space 315

29. Examples of Compact Symmetric Operators 323

29.1 Convolution 323

29.2 The inverse of a differential operator 326

29.3 The inverse of partial differential operators 327

30. Trace Class and Trace Formula 329

30.1 Polar decomposition and singular values 329

30.2 Trace class,trace norm,and trace 330

30.3 The trace formula 334

30.4 The determinant 341

30.5 Examples and counterexamples of trace class operators 342

30.6 The Poisson summation formula 348

30.7 How to express the index of an operator as a difference of traces 349

30.8 The Hilbert-Schmidt class 352

30.9 Determinant and trace for operator in Banach spaces 353

31. Spectral Theory of Symmetric,Normal,and Unitary Operators 354

31.1 The spectrum of symmetric operators 356

31.2 Functional calculus for symmetric operators 358

31.3 Spectral resolution of symmetric operators 361

31.4 Absolutely continuous,singular,and point spectra 364

31.5 The spectral representation of symmetric operators 364

31.6 Spectral resolution of normal operators 370

31.7 Spectral resolution of unitary operators 372

Historical note 375

32. Spectral Theory of Self-Adjoint Operators 377

32.1 Spectral resolution 378

32.2 Spectral resolution using the Cayley transform 389

32.3 A functional calculus for self-adjoint operators 390

33. Examples of Self-Adjoint Operators 394

33.1 The extension of unbounded symmetric operators 394

33.2 Examples of the extension of symmetric operators; deficiency indices 397

33.3 The Friedrichs extension 402

33.4 The Rellich perturbation theorem 406

33.5 The moment problem 410

Historical note 414

34. Semigroups of Operators 416

34.1 Strongly continuous one-parameter semigroups 418

34.2 The generation of semigroups 424

34.3 The approximation of semigroups 427

34.4 Perturbation of semigroups 432

34.5 The spectral theory of semigroups 434

35. Groups of Unitary Operators 440

35.1 Stone's theorem 440

35.2 Ergodic theory 443

35.3 The Koopman group 445

35.4 The wave equation 447

35.5 Translation representation 448

35.6 The Heisenberg commutation relation 455

Historical note 459

36. Examples of Strongly Continuous Semigroups 461

36.1 Semigroups denned by parabolic equations 461

36.2 Semigroups defined by elliptic equations 462

36.3 Exponential decay of semigroups 465

36.4 The Lax-Phillips semigroup 470

36.5 The wave equation in the exterior of an obstacle 472

37. Scattering Theory 477

37.1 Perturbation theory 477

37.2 The wave operators 480

37.3 Existence of the wave operators 482

37.4 The invariance of wave operators 490

37.5 Potential scattering 490

37.6 The scattering operator 491

Historical note 492

37.7 The Lax-Phillips scattering theory 493

37.8 The zeros of the scattering matrix 499

37.9 The automorphic wave equation 500

38. A Theorem of Beurling 513

38.1 The Hardy space 513

38.2 Beurling's theorem 515

38.3 The Titchmarsh convolution theorem 523

Historical note 525

Texts 527

A. Riesz-Kakutani representation theorem 529

A.l Positive linear functionals 529

A.2 Volume 532

A.3 L as a space of functions 535

A.4 Measurable sets and measure 538

A.5 The Lebesgue measure and integral 541

B. Theory of distributions 543

B.l Definitions and examples 543

B.2 Operations on distributions 545

B.3 Local properties of distributions 547

B.4 Applications to partial differential equations 554

B.5 The Fourier transform 558

B.6 Applications of the Fourier transform 568

B.7 Fourier series 569

C. Zorn's Lemma 571

Author Index 573

Subject Index 577