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Main Question or Discussion Point
- Author: Erwin Kreyszig
- Title: Introductory Functional Analysis wih Applications
- Amazon link https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20
- Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous Calculus and Linear algebra.
- Level: Undergrad
Table of Contents:
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[LIST]
[*] Metric Spaces
[LIST]
[*] Metric Space
[*] Further Examples of Metric Spaces
[*] Open Set, Closed Set, Neighborhood
[*] Convergence, Cauchy Sequence, Completeness
[*] Examples. Completeness Proofs
[*] Completion of Metric Spaces
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[*] Normed Spaces. Banach Spaces
[LIST]
[*] Vector Space
[*] Normed Space. Banach Space
[*] Further Properties of Normed Spaces
[*] Finite Dimensional Normed Spaces and Subspaces
[*] Compactness and Finite Dimension
[*] Linear Operators
[*] Bounded and Continuous Linear Operators
[*] Linear Functionals
[*] Linear Operators and Functionals on Finite Dimensional Spaces
[*] Normed Spaces of Operators. Dual Spac
[/LIST]
[*] Inner Produd Spaces. Hilbert Spaces
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[*] Inner Product Space. Hilbert Space
[*] Further Properties of Inner Product Spaces
[*] Orthogonal Complements and Direct Surns
[*] Orthonormal Sets snd Sequences
[*] Series Related to Orthonormal Sequences and Sets
[*] Total Orthonormal Sets and Sequence
[*] Legendre, Hermite and Laguerre Polynomials
[*] Representation of Functionals on Hilbert Spaces
[*] Hilbert-Adjoint Operator
[*] Self-Adjoint, Unitary and Normal Operators
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[*] Fundamental Theorems for Normed and Banach Spaces
[LIST]
[*] Zorn's Lemma
[*] Hahn-Banach Theorem
[*] Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces
[*] Application to Bounded Linear Functionals on [itex]C[a, b][/itex]
[*] Adjoint Operator
[*] Reflexive Spaces
[*] Category Theorem. Uniform Boundedness Theorem
[*] Strong and Weak Convergence
[*] Convergence of Sequences of Operators and Functionals
[*] Application to Summability of Sequences
[*] Numerical Integration and Weak* Convergence
[*] Open Mapping Theorem
[*] Closed Linear Operators. Closed Graph Theorem
[/LIST]
[*] Further Applications: Banach Fixed Point Theorem
[LIST]
[*] Banach Fixed Point Theorem
[*] Application of Banach's Theorem to Linear Equations
[*] Applications of Banach's Theorem to Differential Equations
[*] Application of Banach's Theorem to Integral Equations
[/LIST]
[*] Further Applications: Approximation Theory
[LIST]
[*] Approximation in Normed Spaces
[*] Uniqueness. Strict Convexity
[*] Uniform Approximation
[*] Chebyshev Polynomials
[*] Approximation in Hilbert Space
[*] Splines
[/LIST]
[*] Spectral Theory of Linear Operators in Normed Spaces
[LIST]
[*] Spectral Theory in Finite Dimensional Normed Spaces
[*] Basic Concepts
[*] Spectral Properties of Bounded Linear Operators
[*] Further Properties of Resolvent and Spectrum
[*] Use of Complex Analysis in Spectral Theory
[*] Banach Algebras
[*] Further Properties of Banach Algebras
[/LIST]
[*] Compact Linear Operators on Normed Spaces and Their Spectrum
[LIST]
[*] Compact Linear Operators on Normed Spaces
[*] Further Properties of Compact Linear Operators
[*] Spectral Properties of Compact Linear Operators on Normed Spaces
[*] Further Spectral Properties of Compact Linear Operators
[*] Operator Equations Involving Compact Linear Operators
[*] Further Theorems of Fredholm Type
[*] Fredholm Alternative
[/LIST]
[*] Spectral Theory of Bounded Self-Adjoint Linear Operators
[LIST]
[*] Spectral Properties of Bounded SeIf-Adjoint Linear Operators
[*] Further Spectral Properties of Bounded Self-Adjoint Linear Operators
[*] Positive Operators
[*] Square Roots of a Positive Operator
[*] Projection Operators
[*] Further Properties of Projections
[*] Spectral Family
[*] Spectral Family of a Bounded Self-Adjoint Linear Operator
[*] Spectral Representation of Bounded Self-Adjoint Linear Operators
[*] Extension of the Spectral Theorem to Continuous Functions
[*] Properties of tbe Spectral Family of a Bounded Self-Adjoint Linear Operator
[/LIST]
[*] Unbounded Linear Operators in Hilbert Space
[LIST]
[*] Unbounded Linear Operators and their Hilbert-Adjoint Operators
[*] Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators
[*] Closed Linear Operators and Closures
[*] Spectral Properties of Self-Adjoint Linear Operators
[*] Spectral Representation of Unitary Operators
[*] Spectral Representation of Self-Adjoint Linear Operators
[*] Multiplication Operator and Differentiation Operator
[/LIST]
[*] Unbounded Linear Operaton in Quantum Mechanics
[LIST]
[*] Basic Ideas. States, Observables Position Operator
[*] Momentum Operator. Heisenberg Uncertainty Principle
[*] Time-Independent Schrodinger Equation
[*] Hamilton Operator
[*] Time- Dependent Schrodinger Equation
[/LIST]
[*] Appendix: Some Material for Review and Reference
[LIST]
[*] Sets
[*] Mappings
[*] Families
[*] Equivalence Relations
[*] Compactness
[*] Supremum and Infimum
[*] Cauchy Convergence Criterion
[*] Groups
[/LIST]
[*] Appendix: Answers to Odd-Numbered Problems
[*] Appendix: References
[*] Index
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