Introductory Functional Analysis With Applications

Provides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis. NOTATIONS CHAPTER 1 METRIC SPACES 1.1 Metric Space 1.2 Further Examples of Metric Spaces 1.3 Open Set; Closed Set; Neighborhood 1.4 Convergence. Cauchy Sequence. Completeness 1. 5 Examples. Completeness Proofs I. 6 Completion of Metric Spaces CHAPTER 2 NORMED SPACES. BANACH SPACES Important mncepts; brief orientation about main content Remark on notatioa 2.1 Vector Space 2.2 Normed Space. Banach Space 2.3 Further Properties of Normed Spaces 2.4 Finite Dimensional Normed Spaces and Subspaces 2.5 Compactness and Finite Dimension 2.6 Linear Operators 2.7 Bounded and Continuous Linear Operators 2.8 Linear Functionals 2.9 Linear Operators and Functionals on Finite Dimensional Spaces 2.10 Normed Spaces of Operators. Dual Space CHAPTER 3 INNER PRODUCT SPACES. HILBERT SPACES Important concepts; brief orientation about main content 3.1 Inner Product Space. Hilbert Space 3.2 Further Properties of Inner Product Spaces 3.3 Orthogonal Complements and Direct Sums 3.4 Orthonprmal Sets and Sequences 3.5 Series Related to Orthonogonal Sequences and Sets 3.6 Total Orthonormal Sets and Sequences 3.7 Legendre; Hermite and Laguerre Polynomials 3.8 Representation of Functionals on Hilbert Spaces 3. 9 Hilbert-Adjoint Operator 3.10 Self-Adjoint; Unitary and Nonnal Operators CHAPTER 4 FUNDAMENTAL THEOREMS FOR NORMED AND BANACH SPACES Brief orientation about main content 4.1 Zorn's Lemma 4.2 Hahn-Banach Theorem 4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces 4.4 Application to Bounded Linear Functionals on C[a,b] 4.5 Adjoint Operator 4.6 Reflexive Spaces 4.7 Category Theorem. Uniform Boundedness Theorem 4.8 Strong and Weak Convergence 4.9 Convergence of Sequences of Operators and Functionals 4.10 Application to Summability of Sequences 4.11 Numerical Integration and Weak* Convergence 4.12 Open Mapping Theorem 4.13 Closed Linear Operators. Closed Graph Theorem CHAPTER 5 FURTHER APPLICATIONS: BANACH FIXED POINT THEOREM Brief orientation about main content 5.1 Banach Fixed Point Theorem 5.2 Application of Banach's Theorem to Linear Equations 5.3 Application of Banach's Theorem to Differential Equations 5.4 Application of Banach's Theorem to Integral Equations CHAPTER 6 FURTHER APPLICATIONS: APPROXIMATION THEORY Important concepts,brief orientation about main content 6.1 Approximation in Normed Spaces 6.2 Uniqueness Strict Convexity 6.3 Uniform Approximation 6.4 Chehyshev Polynomials 6.5 Approximation in Hilbert Space 6.6 Splines CHAPTER 7 SPECTRAL THEORY OF LINEAR OPERATORS IN NORMED SPACES Brief orientation about main content of Chap. 7 7.1 Spectral Theory in Finite Dimensional Normed Spaces 7.2 Basic Concepts 7.3 Spectral Properties of Bounded Linear Operators 7.4 Further Properties of Resolvent and Spectrum 7.5 Use of Complex Analysis in Spectral Theory 7.6 Banach Algebras 7.7 Further Properties of Banach Algebras CHAPTER 8 COMPACT LINEAR OPERATORS ON NORMED SPACES AND THEIR SPECTRUM Brief orientation about main content 8.1 Compact Linear Operators on Normed Spaces 8.2 Further Properties of Compact Linear Operators 8.3 Spectral Properties of Compact Linear Operators on Normed Spaces 8.4 Further Spectral Properties of Compact Linear Operators 8.5 Operator Equations lnvolving Compact Linear Operators 8.6 Further Theorems of Fredholm Type 8.7 Fredholm Alternative CHAPTER 9 SPECTRAL THEORY OF BOUNDED SELF-ADJOINT LINEAR OPERATORS Important concepts, brief orientation about main content 9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators 9.2 Further Spectral Properties of Bounded Self -Adjoint Linear Operators 9.3 Positive Operators 9.4 Square Roots of a Positive Operator 9.5 Projection Operators 9.6 Further Properties of Projections 9.7 Speetral Family 9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator 9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators 9.10 Extension of the Spectral Theorem to Continuous Functions 9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator CHAPTER 10 UNBOUNDED LINEAR OPERATORS IN HILBERT SPACE Important concepts, brief orientation about main content 10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators 10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators 10.3 Closed Linear Operators and Closures 10.4 Spectral Properties of Self-AdjointLinear Operators 10.5 Spectral Representation of Unitary Operators 10.6 Spectral Representation of Self-Adjoint Linear Operators 10.7 Multiplication Operator and Differentiation Operator CHAPTER 11 UNBOUNDED LINEAR OPERATORS IN QUANTUM MECHANICS Important concepts, briel orientation about main content 11.1 Basic Ideas. States, Observables, Position Operator 11.2 Momentum Operator. Heisenberg Uncertainty Principle 11.3 Time-Independent Schrödinger Equation 11.4 Hamilton Operator 11.5 Time-Dependent Schrödinger Equation APPENDIX 1 SOME MATERIAL FOR REVIEW AND REFERENCE A1.1 Sets A1.2 Mappings A1.3 Families A1.5 Compactness A1.6 Supremun and lnfimum A1.7 Cauchy Convergence Criterion A1.8 Groups APPENDIX 2 ANSWERS TO ODD NUMBERED PROBLEMS APPENDIX 3 REFERENCES KREYSZIG The Wiley Classics Library consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Emil Artin Geometnc Algebra R. W. Carter Simple Groups Of Lie Type Richard Courant Differential and Integrai Calculus. Volume I Richard Courant Differential and Integral Calculus. Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics. Volume II Harold M. S. Coxeter Introduction to Modern Geometry. Second Edition Charles W. Curtis, Irving Reiner Representation Theory of Finite Groups and Associative Algebras Nelson Dunford, Jacob T. Schwartz unear Operators. Part One. General Theory Nelson Dunford. Jacob T. Schwartz Linear Operators, Part Two. Spectral TheorySelf Adjant Operators in Hilbert Space Nelson Dunford, Jacob T. Schwartz Linear Operators. Part Three. Spectral Operators Peter Henrici Applied and Computational Complex Analysis. Volume IPower Senes-lntegrauon-Contormal Mapping-Locatvon of Zeros Peter Hilton, Yet-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin Kreyszig Introductory Functional Analysis with Applications P. M. Prenter Splines and Variational Methods C. L. Siegel Topics in Complex Function Theory. Volume I Elliptic Functions and Uniformizatton Theory C. L. Siegel Topics in Complex Function Theory. Volume II Automorphic and Abelian Integrals C. L. Siegel Topics In Complex Function Theory. Volume III Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry Erwin Kreyszig. Reprint. Originally Published: New York : Wiley, C1978. Bibliography: P. 675-679.