Linear Operator Theory in Engineering and Science

A unique introduction to the theory of linear operators on Hilbert space.
The author presents the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.

Résumé
The author presents the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.

Contenu
1 Introduction.- 1. Black Boxes.- 2. Structure of the Plane.- 3. Mathematical Modeling.- 4. The Axiomatic Method. The Process of Abstraction.- 5. Proofs of Theorems.- 2 Set-Theoretic Structure.- 1. Introduction.- 2. Basic Set Operations.- 3. Cartesian Products.- 4. Sets of Numbers.- 5. Equivalence Relations and Partitions.- 6. Functions.- 7. Inverses.- 8. Systems Types.- 3 Topological Structure.- 1. Introduction.- A Introduction to Metric Spaces.- 2. Metric Spaces: Definition.- 3. Examples of Metric Spaces.- 4. Subspaces and Product Spaces.- 5. Continuous Functions.- 6. Convergent Sequences.- 7. A Connection Between Continuity and Convergence.- B Some Deeper Metric Space Concepts.- 8. Local Neighborhoods.- 9. Open Sets.- 10. More on Open Sets.- 11. Examples of Homeomorphic Metric Spaces.- 12. Closed Sets and the Closure Operation.- 13. Completeness.- 14. Completion of Metric Spaces.- 15. Contraction Mapping.- 16. Total Boundedness and Approximations.- 17. Compactness.- 4 Algebraic Structure.- 1. Introduction.- A Introduction to Linear Spaces.- 2. Linear Spaces and Linear Subspaces.- 3. Linear Transformations.- 4. Inverse Transformations.- 5. Isomorphisms.- 6. Linear Independence and Dependence.- 7. Hamel Bases and Dimension.- 8. The Use of Matrices to Represent Linear Transformations.- 9. Equivalent Linear Transformations.- B Further Topics.- 10. Direct Sums and Sums.- 11. Projections.- 12. Linear Functionals and the Algebraic Conjugate of a Linear Space.- 13. Transpose of a Linear Transformation.- 5 Combined Topological and Algebraic Structure.- 1. Introduction.- A Banach Spaces.- 2. Definitions.- 3. Examples of Normal Linear Spaces.- 4. Sequences and Series.- 5. Linear Subspaces.- 6. Continuous Linear Transformations.- 7. Inverses and Continuous Inverses.- 8. Operator Topologies.- 9. Equivalence of Normed Linear Spaces.- 10. Finite-Dimensional Spaces.- 11. Normed Conjugate Space and Conjugate Operator.- B Hilbert Spaces.- 12. Inner Product and Hilbert Spaces.- 13.Examples.- 14. Orthogonality.- 15. Orthogonal Complements and the Projection Theorem.- 16. Orthogonal Projections.- 17. Orthogonal Sets and Bases: Generalized Fourier Series.- 18. Examples of Orthonormal Bases.- 19. Unitary Operators and Equivalent Inner Product Spaces.- 20. Sums and Direct Sums of Hilbert Spaces.- 21. Continuous Linear Functionals.- C Special Operators.- 22. The Adjoint Operator.- 23. Normal and Self-Adjoint Operators.- 24. Compact Operators.- 25. Foundations of Quantum Mechanics.- 6 Analysis of Linear Operators (Compact Case).- 1. Introduction.- A An Illustrative Example.- 2. Geometric Analysis of Operators.- 3. Geometric Analysis. The Eigenvalue-Eigenvector Problem.- >4. A Finite-Dimensional Problem.- B The Spectrum.- 5. The Spectrum of Linear Transformations.- 6. Examples of Spectra.- 7. Properties of the Spectrum.- C Spectral Analysis.- 8. Resolutions of the Identity.- 9. Weighted Sums of Projections.- 10. Spectral Properties of Compact, Normal, and Self-AdjointOperators.- 11. The Spectral Theorem.- 12. Functions of Operators (Operational Calculus).- 13. Applications of the Spectral Theorem.- 14. Nonnormal Operators.- 7 Analysis of Unbounded Operators.- 1. Introduction.- 2. Green's Functions.- 3. Symmetric Operators.- 4. Examples of Symmetric Operators.- 5. Sturm-Liouville Operators.- 6. Gårding's Inequality.- 7. Elliptic Partial Differential Operators.- 8. The Dirichlet Problem.- 9. The Heat Equation and Wave Equation.- 10. Self-Adjoint Operators.- 11. The Cayley Transform.- 12. Quantum Mechanics, Revisited.- 13. Heisenberg Uncertainty Principle.- 14. The Harmonic Oscillator.- Appendix A The Hölder, Schwartz, and Minkowski Inequalities.- Appendix B Cardinality.- Appendix C Zorn's Lemma.- Appendix D Integration and Measure Theory.- 1. Introduction.- 2. The Riemann Integral.- 3. A Problem with the Riemann Integral.- 5. Null Sets.- 6. Convergence Almost Everywhere.- 7. The Lebesgue Integral.- 8. Limit Theorems.- 9. Miscellany.- 10. Other Definitions of the Integral.- 13. Differentiation.- 14. The Radon-Nikodym Theorem.- 15. Fubini Theorem.- Appendix E Probability Spaces and Stochastic Processes.- 1. Probability Spaces.- 2. Random Variables and Probability Distributions.- 3. Expectation.- 4. Stochastic Independence.- 5. Conditional Expectation Operator.- 6. Stochastic Processes.- Index of Symbols.