Verfügbarkeit: Functional analysis | Technische Universität München

Functional analysis:

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Bibliographische Detailangaben
Beteilige Person:	Lax, Peter D. 1926- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York Wiley-Interscience 2002
Schriftenreihe:	Pure and applied mathematics
Schlagwörter:	Funktionalanalysis Lehrbuch
Links:	http://www.loc.gov/catdir/toc/wiley021/2001046547.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009878719&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Hier auch später erschienene, unveränderte Nachdrucke
Umfang:	XIX, 580 Seiten
ISBN:	0471556041 9780471556046

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adam_text	CONTENTS Foreword xvii 1. Linear Spaces 1 Axioms for linear spaces — Infinite-dimensional examples — Subspace, linear span —Quotient space—Isomorphism — Convex sets — Extreme subsets 2. Linear Maps 8 2.1 Algebra of linear maps, 8 Axioms for linear maps — Sums and composites — Invertible linear maps —Nullspace and range — Invariant subspaces 2.2. Index of a linear map, 12 Degenerate maps — Pseudoinverse — Index — Product formula for the index — Stability of the index 3. The Hahn-Banach Theorem 19 3.1 The extension theorem, 19 Positive homogeneous, subadditive functionals— Extension of linear functionals — Gauge functions of convex sets 3.2 Geometric Hahn-Banach theorem, 21 The hyperplane separation theorem 3.3 Extensions of the Hahn-Banach theorem, 24 The Agnew-Morse theorem — The Bohnenblust-Sobczyk-Soukhomlinov theorem 4. Applications of the Hahn-Banach theorem 29 4.1 Extension of positive linear functionals, 29 4.2 Banach limits, 31 ,¡ CONTENTS 4.3 Finitely additive invariant set functions, 33 Historical note, 34 5. Normed Linear Spaces 36 5.1 Norms, 36 Norms for quotient spaces — Complete normed linear spaces — The spaces С, В — Lp spaces and Holder s inequality — Sobolev spaces, embedding theorems — Separable spaces 5.2 Noncompactness of the unit ball, 43 Uniform convexity — The Mazur-Ulam theorem on isometries 5.3 Isometries, 47 6. Hubert Space 52 6.1 Scalar product, 52 Schwarz inequality — Parallelogram identity — Completeness, closure — Ѓ2, L2 6.2 Closest point in a closed convex subset, 54 Orthogonal complement of a subspace — Orthogonal decomposition 6.3 Linear functionals, 56 The Riesz-Frechet representation theorem — Lax-Milgram lemma 6.4 Linear span, 58 Orthogonal projection — Orthonormal bases, Gram-Schmidt process — Isometries of a Hubert space 7. Applications of Hilbert Space Results 63 7.1 Radon-Nikodym theorem, 63 7.2 Dirichlet s problem, 65 Use of the Riesz-Frechet theorem — Use of the Lax-Milgram theorem — Use of orthogonal decomposition 8. Duals of Normed Linear Spaces 72 8.1 Bounded linear functionals, 72 Dual space 8.2 Extension of bounded linear functionals, 74 Dual characterization of norm — Dual characterization of distance from a subspace — Dual characterization of the closed linear span of a set CONTENTS vii 8.3 Reflexive spaces, 78 Reflexivity of Lp, I < ρ < ею — Separable spaces — Separability of the dual — Dual of C(Q), Q compact — Reflexivity of subspaces 8.4 Support function of a set, 83 Dual characterization of convex hull — Dual characterization of distance from a closed, convex set 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 Principle of uniform boundedness — Weakly sequentially closed convex sets 10.2 Weak sequential compactness, 104 Compactness of unit ball in reflexive space 10.3 Weak* convergence, 105 Helly s theorem 11. Applications of Weak Convergence 108 11.1 Approximation of the 8 function by continuous functions, 108 Toeplitz s theorem on summability 11.2 Divergence of Fourier series, 109 11.3 Approximate quadrature, 110 11.4 Weak and strong analyticky of vector-valued functions, 111 11.5 Existence of solutions of partial differential equations, 112 Galerkin s method 11.6 The representation of analytic functions with positive real part, 115 Herglotz-Riesz theorem 12. The Weak and Weak* Topologies 118 Comparison with weak sequential topology — Closed convex sets in the weak topology — Weak compactness — Alaoglu s theorem viii CONTENTS 13. Locally Convex Topologies and the Krein-MÜman Theorem 122 13.1 Separation of points by linear functionaLs, 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 Carathéodory 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 Norm of a bounded linear map — Transpose 15.2 Strong and weak topologies, 165 Strong and weak sequential convergence 15.3 Principle of uniform boundedness, 166 15.4 Composition of bounded maps, 167 15.5 The open mapping principle, 168 Closed graph theorem Historical note, 172 16. Examples of Bounded Linear Maps 173 16.1 Boundedness of integral operators, 173 Integral operators of Hilbert-Schmidt type — Integral operators of Holmgren type 16.2 The convexity theorem of Marcel Riesz, 177 16.3 Examples of bounded integral operators, 180 The Fourier transform, Parseval s theorem and Hausdorff-Young inequality — The Hubert transform — The Laplace transform — The Hilbert-Hankel transform CONTENTS ix 16.4 Solution Operators for hyperbolic equations, ] 86 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 Invertible elements — Resolvent set and spectrum — Resolvent — Spectral radius 17.2 Functional calculus, 197 Spectral mapping theorem — Projections 18. Gelfand s Theory of Commutative Banach Algebras 202 Homomorphisms into С — Maximal ideals — Mazur s lemma — The spectrum as the range of homomorphisms — The spectral mapping theorem revisited — The Gelfand representation — Gelfand topology 19. Applications of Gelfand s Theory of Commutative Banach Algebras 2 1 0 19.1 The algebra Ш), 210 19.2 Gelfand compactification, 210 19.3 Absolutely convergent Fourier series, 212 19.4 Analytic functions in the closed unit disk, 213 Analytic functions in the closed polydisk 19.5 Analytic functions in the open unit disk, 214 19.6 Wiener s Tauberian theorem. 215 19.7 Commutative ^-algebras. 221 Historical note, 224 20. Examples of Operators and Their Spectra 226 20. 1 Invertible maps. 226 Boundary points of the spectrum 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 Compact maps form a two-sided ideal — Identity plus compact map has index zero χ CONTENTS 21.2 The spectral theory of compact maps, 238 The transpose of a compact operator is compact — The Fredholm alternative Historical note, 244 22. Examples of Compact Operators 245 22.1 Compactness criteria, 245 Arzela-Ascoli compactness criterion — Rellich compactness criterion 22.2 Integral operators, 246 Hilbert-Schmidt operators 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 Invariant probability density 23.3 Inverse of a second order elliptic operator, 258 24. Fredholm s Theo ryof Integral Equations 260 24.1 The Fredholm determinant and the Fredholm resolvent, 260 The spectrum of Fredholm operators — A trace formula for Fredholm operators 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 The von Neumann-Aronszajn-Smith theorem 25.2 Nested invariant subspaces, 277 Ringrose s theorem — Unicellular operators: the Brodsky-Donoghue theorem — The Robinson-Bernstein and Lomonsov theorems — Enflons example 26. Harmonic Analysis on a Halfline 284 26.1 The Phragmén-Lindelöf principle for harmonic functions, 284 CONTENTS xi 26.2 An abstract Pragmén-Lindelöf principle, 285 Interior compactness 26.3 Asymptotic expansion, 297 Solutions of elliptic differential equations in a half-cylinder 27. Index Theory 300 27.1 The Noether index, 301 Pseudoinverse — Stability of index — Product formula — Hörmander s stability theorem Historical note, 305 27.2 Toeplitz operators, 305 Index-winding number — The inversion of Toeplitz operators — Discontinuous symbols — Matrix Toeplitz operators 27.3 Hankel operators, 312 28. Compact Symmetric Operators in Hubert Space 315 Variational principle for eigenvalues — Completeness of eigenfunctions — The variational principles of Fisher and Courant — Functional calculus — Spectral theory of compact normal operators — Unitary operators 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 Matrix trace 30.3 The trace formula, 334 Weyľs inequalities — Lidskii s theorem 30.4 The determinant, 341 30.5 Examples and counterexamples of trace class operators, 342 Mercer s theorem — The trace of integral operators — A Volteira integral operator — The trace of the powers of an operator xi¡ CONTENTS 30.6 The Poisson summation formula, 348 Convolution on Sl and the convergence of Fourier series — The Selberg trace formula 30.7 How to express the index of an operator as a difference of traces, 349 30.8 The Hilbert-Schmidt class, 352 Relation of Hilbert-Schmidt class and trace class 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 Reality of spectrum — Upper and lower bounds for the spectrum — Spectral radius 31.2 Functional calculus for symmetric operators, 358 The square root of a positive operator — Polar decomposition of bounded operators 31.3 Spectral resolution of symmetric operators, 361 Projection-valued measures 31.4 Absolutely continuous, singular, and point spectra, 364 31.5 The spectral representation of symmetric operators, 364 Spectral multiplicity — Unitary equivalence 31.6 Spectral resolution of normal operators, 370 Functional calculus — Commutative B* -algebras 31.7 Spectral resolution of unitary operators, 372 Historical note, 375 32. Spectral Theory of Self-Adjoint Operators 377 The Hellinger-Toeplitz theorem — Definition of self-adjointness — Domain 32.1 Spectral resolution, 378 Sharpening of Herglotz s theorem — Cauchy transform of measures — The spectrum of a self-adjoint operator — Representation of the resolvent as a Cauchy transform — Projection-valued measures 32.2 Spectral resolution using the Cayley transform, 389 32.3 A functional calculus for self-adjoint operators, 390 CONTENTS xiii 33. Examples of Self-Adjoint Operators 394 33.1 The extension of unbounded symmetric operators, 394 Closure of a symmetric operator 33.2 Examples of the extension of symmetric operators; deficiency indices, 397 The operator i(d/dx) on C¿(M), С( ,(К+), and C¿(0, 1)— Deficiency indices and von Neumann s theorem — Symmetric operators in a real Hubert space 33.3 The Friedrichs extension, 402 Semibounded symmetric operators — Symmetric ODE — Symmetric elliptic PDE 33.4 The Rellich perturbation theorem, 406 Self-adjointness of Schrödinger operators with singular potentials 33.5 The moment problem, 410 The Hamburger and Stieltjes moment problems — Uniqueness, or not, of the moment problem Historical note, 414 34. Semigroups of Operators 416 34.1 Strongly continuous one-parameter semigroups, 418 Infinitesimal generator — Resolvent — Laplace transform 34.2 The generation of semigroups, 424 The Hille-Yosida theorem 34.3 The approximation of semigroups, 427 The Lax equivalence theorem — Trotter s product formula — Strang s product formula 34.4 Perturbation of semigroups, 432 Lumer-Phillip s theorem — Trotter s perturbation theorem 34.5 The spectral theory of semigroups, 434 Phillip s spectral mapping theorem — Adjoint semigroups — Semigroups of eventually compact operators 35. Groups of Unitary Operators 440 35.1 Stone s theorem, 440 Generation of unitary groups — Positive definiteness and Bochner s theorem 35.2 Ergodic theory, 443 von Neumann s mean ergodic theorem xiv CONTENTS 35.3 The Koopman group, 445 Volume-preserving flows — Metric transitivity — Time average — Space average 35.4 The wave equation, 447 In full space-time — In the exterior of an obstacle 35.5 Translation representation, 448 Sinai s theorem — Incoming subspaces — Solution of wave equation in odd number of space dimensions — Wave propagation outside an obstacle 35.6 The Heisenberg commutation relation, 455 The uncertainty principle — Weyl s form of the commutation relation —von Neumman s theorem on pairs of operators that satisfy the commutation relation Historical note, 459 36. Examples of Strongly Continuous Semigroups 461 36.1 Semigroups defined 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 Faddeev and Pavlor s theory — The Riemann hypothesis 38. A Theorem of Beurling 513 38.1 The Hardy space, 513 38.2 Beurling s theorem, 515 Inner and outer factors — Factorization in the algebra of bounded analytic functions CONTENTS xv 38.3 The Titchmarsh convolution theorem, 523 Historical note, 525 Texts 527 A. Riesz-Kakutani representation theorem 529 A. 1 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.I Definitions and examples, 543 B.2 Operations on distributions, 544 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
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title	Functional analysis
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title_full	Functional analysis Peter D. Lax
title_fullStr	Functional analysis Peter D. Lax
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