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Wiley classics library edition published 1989

New York : John Wiley & Sons, c1989

xiv, 688 s. : il. ; 23 cm

ISBN 0-471-50459-9 (brož.)

Wiley classics library

Obsahuje bibliografii na s. 675-479 a rejstřík

000126669

Chapter 1. Metric Spaces ...1 // 1.1 Metric Space 2 // 1.2 Further Examples of Metric Spaces 9 // 1.3 Open Set, Closed Set, Neighborhood 17 // 1.4 Convergence, Cauchy Sequence, Completeness 25 // 1.5 Examples. Completeness Proofs 32 // 1.6 Completion of Metric Spaces 41 // Chapter 2. Normed Spaces. Banach Spaces ...49 // 2.1 Vector Space 50 // 2.2 Normed Space. Banach Space 58 // 2.3 Further Properties of Normed Spaces 67 // 2.4 Finite Dimensional Normed Spaces and Subspaces 72 // 2.5 Compactness and Finite Dimension 77 // 2.6 Linear Operators 82 // 2.7 Bounded and Continuous Linear Operators 91 // 2.8 Linear Functionals 103 // 2.9 Linear Operators and Functionals on Finite Dimensional Spaces 111 // 2.10 Normed Spaces of Operators. Dual Space 117 // Chapter 3. Inner Product Spaces. Hilbert Spaces ...127 // 3.1 Inner Product Space. Hilbert Space 128 // 3.2 Further Properties of Inner Product Spaces 136 // 3.3 Orthogonal Complements and Direct Sums 142 // 3.4 Orthonormal Sets and Sequences 151 // 3.5 Series Related to Orthonormal Sequences and Sets 160 // 3.6 Total Orthonormal Sets and Sequences 167 // 3.7 Legendre, Hermité and Laguerre Polynomials 175 // 3.8 Representation of Functionals on Hilbert Spaces 188 // 3.9 Hilbert-Adjoint Operator 195 // 3.10 Self-Adjoint, Unitary and Normal Operators 201 // Chapter 4. Fundamental Theorems for Normed and Banach Spaces ...209 // 4.1 Zorn’s Lemma 210 // 4.2 Hahn-Banach Theorem 213 // 4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces 218 // 4.4 Application to Bounded Linear Functionals on C[fl,b] 225 // 4.5 Adjoint Operator 231 // 4.6 Reflexive Spaces 239 // 4.7 Category Theorem. Uniform Boundedness Theorem 246 // 4.8 Strong and Weak Convergence 256 // 4.9 Convergence of Sequences of Operators and Functionals 263 // 4.10 Application to Summability of Sequences 269 // 4.11 Numerical Integration and Weak* Convergence 276 //

4.12 Open Mapping Theorem 285 // 4.13 Closed Linear Operators. Closed Graph Theorem 291 // Chapter 5. Further Applications: Banach Fixed Point Theorem ... 299 // 5.1 Banach Fixed Point Theorem 299 // 5.2 Application of Banach’s Theorem to Linear Equations 307 // 5.3 Applications of Banach’s Theorem to Differential Equations 314 // 5.4 Application of Banach’s Theorem to Integral Equations 319 // Chapter 6. Further Applications: Approximation Theory ...327 // 6.1 Approximation in Normed Spaces 327 // 6.2 Uniqueness, Strict Convexity 330 // 6.3 Uniform Approximation 336 // 6.4 Chebyshev Polynomials 345 // 6.5 Approximation in Hilbert Space 352 // 6.6 Splines 356 // Chapter 7. Spectral Theory of Linear Operators in Normed Spaces ...363 // 7.1 Spectral Theory in Finite Dimensional Normed Spaces 364 // 7.2 Basic Concepts 370 // 7.3 Spectral Properties of Bounded Linear Operators 374 // 7.4 Further Properties of Resolvent and Spectrum 379 // 7.5 Use of Complex Analysis in Spectral Theory 386 // 7.6 Banach Algebras 394 // 7.7 Further Properties of Banach Algebras 398 // Chapter 8. Compact Linear Operators on Normed Spaces and Their Spectrum ...405 // 8.1 Compact Linear Operators on Normed Spaces 405 // 8.2 Further Properties of Compact Linear Operators 412 // 8.3 Spectral Properties of Compact Linear Operators on Normed Spaces 419 // 8.4 Further Spectral Properties of Compact Linear Operators 428 // 8.5 Operator Equations Involving Compact Linear Operators 436 // 8.6 Further Theorems of Fredholm Type 442 // 8.7 Fredholm Alternative 451 // Chapter 9. Spectral Theory of Bounded Self-Adjoint Linear Operators ...459 // 9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators 460 // 9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators 465 // 9.3 Positive Operators 469 // 9.4 Square Roots of a Positive Operator 476 // 9.5 Projection Operators 480 // 9.6 Further Properties of Projections 486 // 9.7 Spectral Family 492 //

9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator 497 // 9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators 505 // 9.10 Extension of the Spectral Theorem to Continuous Functions 512 // 9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator 516 // Chapter 10. Unbounded Linear Operators in Hilbert Space ...523 // 10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators 524 // 10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators 530 // 10.3 Closed Linear Operators and Closures 535 // 10.4 Spectral Properties of Self-Adjoint Linear Operators 541 // 10.5 Spectral Representation of Unitary Operators 546 // 10.6 Spectral Representation of Self-Adjoint Linear Operators 556 // 10.7 Multiplication Operator and Differentiation Operator 562 // Chapter 11. Unbounded Linear Operators in Quantum Mechanics ...571 // 11.1 Basic Ideas. States, Observables, Position Operator 572 // 11.2 Momentum Operator. Heisenberg Uncertainty Principle 576 // 11.3 Time-Independent Schrödinger Equation 583 // 11.4 Hamilton Operator 590 // 11.5 Time-Dependent Schrödinger Equation 598 // Appendix 1. Some Material for Review and Reference ...609 // A 1.1 Sets 609 // A 1.2 Mappings 613 // A 1.3 Families 617 // A 1.4 Equivalence Relations 618 // A 1.5 Compactness 618 // A 1.6 Supremum and Infimum 619 // A 1.7 Cauchy Convergence Criterion 620 // A 1.8 Groups 622 // Appendix 2. Answers to Odd-Numbered Problems ...623 // Appendix 3. References ...675 // Index ...681