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.
Author(s): Erwin Kreyszig
Publisher: John Wiley & Sons Inc
Year: 1978
Language: English
Pages: XVI, 688
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
