Linear Operator Theory in Engineering and Science

by Naylor, Arch W.; Sell, George R.

• ISBN13: 9780387950013
• ISBN10: 038795001X
• Edition: Reprint
• Format: Paperback
• Copyright: 2000-01-01
• Publisher: Springer Verlag

Summary

This book is a unique introduction to the theory of linear operators on Hilbert space. The authors' goal is to present 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. First published in 1971, Linear Operator inEngineering and Scienceshas since proved to be a popular and very useful textbook.

Table of Contents

Preface	vii
Introduction	1	(10)
Black Boxes	2	(2)
Structure of the Plane	4	(1)
Mathematical Modeling	5	(1)
The Axiomatic Method. The Process of Abstraction	6	(1)
Proofs of Theorems	7	(4)
Set-Theoretic Structure	11	(32)
Introduction	12	(2)
Basic Set Operations	14	(3)
Cartesian Products	17	(1)
Sets of Numbers	18	(1)
Equivalence Relations and Partitions	19	(3)
Functions	22	(7)
Inverses	29	(9)
Systems Types	38	(5)
Topological Structure	43	(116)
Introduction	44	(1)
Introduction to Metric Spaces	45	(1)
Metric Spaces: Definition	45	(2)
Examples of Metric Spaces	47	(9)
Subspaces and Product Spaces	56	(5)
Continuous Functions	61	(8)
Convergent Sequences	69	(5)
A Connection Between Continuity and Convergence	74	(3)
Some Deeper Metric Space Concepts	77	(1)
Local Neighborhoods	77	(5)
Open Sets	82	(10)
More on Open Sets	92	(5)
Examples of Homeomorphic Metric Spaces	97	(4)
Closed Sets and the Closure Operations	101	(11)
Completeness	112	(8)
Completion of Metric Spaces	120	(5)
Contraction Mapping	125	(9)
Total Boundedness and Approximations	134	(7)
Compactness	141	(18)
Algebraic Structure	159	(54)
Introduction	160	(1)
Introduction to Linear Spaces	161	(1)
Linear Spaces and Linear Subspaces	161	(4)
Linear Transformations	165	(6)
Inverse Transformations	171	(2)
Isomorphisms	173	(3)
Linear Independence and Dependence	176	(7)
Hamel Bases and Dimension	183	(5)
The Use of Matrices to Represent Linear Transformations	188	(4)
Equivalent Linear Transformations	192	(4)
Further Topics	196	(1)
Direct Sums and Sums	196	(5)
Projections	201	(3)
Linear Functionals and the Algebraic Conjugate of a Linear Space	204	(4)
Transpose of a Linear Transformation	208	(5)
Combined Topological and Algebraic Structure	213	(182)
Introduction	214	(1)
Banach Spaces	215	(1)
Definitions	215	(3)
Examples of Normal Linear Spaces	218	(6)
Sequences and Series	224	(5)
Linear Subspaces	229	(5)
Continuous Linear Transformations	234	(9)
Inverses and Continuous Inverses	243	(4)
Operator Topologies	247	(10)
Equivalence of Normed Linear Spaces	257	(7)
Finite-Dimensional Spaces	264	(6)
Normed Conjugate Space and Conjugate Operator	270	(2)
Hilbert Spaces	272	(1)
Inner Product and Hilbert Spaces	272	(6)
Examples	278	(4)
Orthogonality	282	(10)
Orthogonal Complements and the Projection Theorem	292	(8)
Orthogonal Projections	300	(5)
Orthogonal Sets and Bases: Generalized Fourier Series	305	(17)
Examples of Orthonormal Bases	322	(9)
Unitary Operators and Equivalent Inner Product Spaces	331	(9)
Sums and Direct Sums of Hilbert Spaces	340	(4)
Continuous Linear Functionals	344	(8)
Special Operators	352	(1)
The Adjoint Operator	352	(15)
Normal and Self-Adjoint Operators	367	(12)
Compact Operators	379	(9)
Foundations of Quantum Mechanics	388	(7)
Analysis of Linear Operators (Compact Case)	395	(90)
Introduction	396	(1)
An Illustrative Example	397	(1)
Geometric Analysis of Operators	397	(2)
Geometric Analysis. The Eigenvalue-Eigenvector Problem	399	(2)
A Finite-Dimensional Problem	401	(10)
The Spectrum	411	(1)
The Spectrum of Linear Transformations	411	(3)
Examples of Spectra	414	(17)
Properties of the Spectrum	431	(8)
Spectral Analysis	439	(1)
Resolutions of the Identity	439	(10)
Weighted Sums of Projections	449	(1)
Spectral Properties of Compact, Normal, and Self-Adjoint Operators	449	(10)
The Spectral Theorem	459	(9)
Functions of Operators (Operational Calculus)	468	(2)
Applications of the Spectral Theorem	470	(6)
Nonnormal Operators	476	(9)
Analysis of Unbounded Operators	485	(63)
Introduction	486	(2)
Green's Functions	488	(5)
Symmetric Operators	493	(2)
Examples of Symmetric Operators	495	(3)
Sturm-Liouville Operators	498	(7)
Garding's Inequality	505	(5)
Elliptic Partial Differential Operators	510	(6)
The Dirichlet Problem	516	(7)
The Heat Equations and Wave Equation	523	(4)
Self-Adjoint Operators	527	(6)
The Cayley Transform	533	(6)
Quantum Mechanics, Revisited	539	(2)
Heisenberg Uncertainty Principle	541	(2)
The Harmonic Oscillator	543	(5)
Appendix A The Holder, Schwartz, and Minkowski Inequalities	548	(4)
Appendix B Cardinality	552	(4)
Appendix C Zorn's Lemma	556	(2)
Appendix D Integration and Measure Theory	558	(41)
1. Introduction	558	(1)
2. The Riemann Integral	559	(5)
3. A Problem with the Riemann Integral	564	(1)
4. The Space C0	564	(2)
5. Null Sets	566	(3)
6. Convergence Almost Everywhere	569	(3)
7. The Lebesgue Integral	572	(4)
8. Limit Theorems	576	(5)
9. Miscellany	581	(5)
10. Other Definitions of the Integral	586	(3)
11. The Lebesgue Space, Lp	589	(2)
12. Dense Subspaces of Lp, 1 ≤ p < ∞	591	(2)
13. Differentiation	593	(3)
14. The Radon-Nikodym Theorem	596	(2)
15. Fubini Theorem	598	(1)
Appendix E Probability Spaces and Stochastic Processes	599	(16)
1. Probability Spaces	599	(1)
2. Random Variables and Probability Distributions	600	(2)
3. Expectation	602	(1)
4. Stochastic Independence	603	(1)
5. Conditional Expectation Operator	604	(3)
6. Stochastic Processes	607	(8)
Index of Symbols	615	(2)
Index	617