Introductory functional analysis with applications

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