Analysis Introductory Functional Analysis with Applications by Kreyszig

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"Introductory Functional Analysis with Applications" by Erwin Kreyszig is a comprehensive undergraduate textbook that covers essential topics in functional analysis while minimizing the complexity of measure theory and topology. The book begins with foundational concepts such as metric spaces, normed spaces, and Banach spaces, progressing to inner product spaces and Hilbert spaces. Key discussions include the properties of linear operators, bounded and continuous linear operators, and the significance of fundamental theorems like Zorn's Lemma and the Hahn-Banach Theorem. Applications of the Banach Fixed Point Theorem to linear and differential equations are highlighted, alongside approximation theory and spectral theory of linear operators. The text also addresses compact linear operators and their spectral properties, as well as unbounded linear operators in Hilbert space, with a specific focus on quantum mechanics concepts such as the Schrödinger equation. The inclusion of numerous problems enhances the learning experience, making it particularly suitable for students in physics.

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  • 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:
Code:
[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
[/LIST]
[*] 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
[LIST]
[*] 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
[/LIST]
[*] 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
[/LIST]
 
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This book is great. Measure theory & topology is kept to a minimum, and there's a chapter on quantum mechanics at the end, which would probably make it better for physics than math. oh, & 900 problems too.
 
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The book is fascinating. If your education includes a typical math degree curriculum, with Lebesgue integration, functional analysis, etc, it teaches QFT with only a passing acquaintance of ordinary QM you would get at HS. However, I would read Lenny Susskind's book on QM first. Purchased a copy straight away, but it will not arrive until the end of December; however, Scribd has a PDF I am now studying. The first part introduces distribution theory (and other related concepts), which...
I've gone through the Standard turbulence textbooks such as Pope's Turbulent Flows and Wilcox' Turbulent modelling for CFD which mostly Covers RANS and the closure models. I want to jump more into DNS but most of the work i've been able to come across is too "practical" and not much explanation of the theory behind it. I wonder if there is a book that takes a theoretical approach to Turbulence starting from the full Navier Stokes Equations and developing from there, instead of jumping from...

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