# Introductory Functional Analysis with Applications by Kreyszig

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## Main Question or Discussion Point

• Author: Erwin Kreyszig
• Title: Introductory Functional Analysis wih Applications
• Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous Calculus and Linear algebra.

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[*] Metric Spaces
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[*] 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
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[*] 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
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[*] 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
[*] Self-Adjoint, Unitary and Normal Operators
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[*] Fundamental Theorems for Normed and Banach Spaces
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[*] Zorn's Lemma
[*] Hahn-Banach Theorem
[*] Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces
[*] Application to Bounded Linear Functionals on $C[a, b]$
[*] 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
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[*] Further Applications: Banach Fixed Point Theorem
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[*] 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
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[*] Further Applications: Approximation Theory
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[*] Approximation in Normed Spaces
[*] Uniqueness. Strict Convexity
[*] Uniform Approximation
[*] Chebyshev Polynomials
[*] Approximation in Hilbert Space
[*] Splines
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[*] Spectral Theory of Linear Operators in Normed Spaces
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[*] 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
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[*] Compact Linear Operators on Normed Spaces and Their Spectrum
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[*] 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
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[*] Spectral Theory of Bounded Self-Adjoint Linear Operators
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[*] 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
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[*] Unbounded Linear Operators in Hilbert Space
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[*] Unbounded Linear Operators and their Hilbert-Adjoint 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
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[*] Unbounded Linear Operaton in Quantum Mechanics
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[*] Basic Ideas. States, Observables Position Operator
[*] Momentum Operator. Heisenberg Uncertainty Principle
[*] Time-Independent Schrodinger Equation
[*] Hamilton Operator
[*] Time- Dependent Schrodinger Equation
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[*] Appendix: Some Material for Review and Reference
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[*] Sets
[*] Mappings
[*] Families
[*] Equivalence Relations
[*] Compactness
[*] Supremum and Infimum
[*] Cauchy Convergence Criterion
[*] Groups
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[*] Appendix: Answers to Odd-Numbered Problems
[*] Appendix: References
[*] Index
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• Jianphys17

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