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Main Question or Discussion Point
- Author: Serge Lang
- Title: Undergraduate Analysis
- Amazon Link: https://www.amazon.com/dp/1441928537/?tag=pfamazon01-20
- Prerequisities: Calculus, Proofs
- Level: Undergrad
Table of Contents:
Code:
[LIST]
[*] Foreword
[*] Review of Calculus
[LIST]
[*] Sets and Mappings
[LIST]
[*] Sets
[*] Mappings
[*] Natural Numbers and Induction
[*] Denumerable Sets
[*] Equivalence Relations
[/LIST]
[*] Real Numbers
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[*] Algebraic Axioms
[*] Ordering Axioms
[*] Integers and Rational Numbers
[*] The Completeness Axiom
[/LIST]
[*] Limits and Continuous Functions
[LIST]
[*] Sequences of Numbers
[*] Functions and Limits
[*] Limits with Infinity
[*] Continuous Functions
[/LIST]
[*] Differentiation
[LIST]
[*] Properties of the Derivative
[*] Mean Value Theorem
[*] Inverse Functions
[/LIST]
[*] Elementary Functions
[LIST]
[*] Exponential
[*] Logarithm
[*] Sine and Cosine
[*] Complex Numbers
[/LIST]
[*] The Elementary Real Integral
[LIST]
[*] Characterization of the Integral
[*] Properties of the Integral
[*] Taylor's Formula
[*] Asymptotic Estimates and Stirling's Formula
[/LIST]
[/LIST]
[*] Convergence
[LIST]
[*] Normed Vector Spaces
[LIST]
[*] Vector Spaces
[*] Normed Vector Spaces
[*] n-Space and Function Spaces
[*] Completeness
[*] Open and Closed Sets
[/LIST]
[*] Limits
[LIST]
[*] Basic Properties
[*] Continuous Maps
[*] Limits in Function Spaces
[*] Completion of a Normed Vector Space
[/LIST]
[*] Compactness
[LIST]
[*] Basic Properties of Compact Sets
[*] Continuous Maps on Compact Sets
[*] Algebraic Closure of the Complex Numbers
[*] Relation with Open Coverings
[/LIST]
[*] Series
[LIST]
[*] Basic Definitions
[*] Series of Positive Numbers
[*] Non-Absolute Convergence
[*] Absolute Convergence in Vector Spaces
[*] Absolute and Uniform Convergence
[*] Power Series
[*] Differentiation and Integration of Series
[/LIST]
[*] The Integral in One Variable
[LIST]
[*] Extension Theorem for Linear Maps
[*] Integral of Step Maps
[*] Approximation by Step Maps
[*] Properties of the Integral
[*] Appendix: The Lebesgue Integral
[*] The Derivative
[*] Relation Between the Integral and the Derivative
[*] Interchanging Derivatives and Integrals
[/LIST]
[/LIST]
[*] Applications of the Integral
[LIST]
[*] Approximation with Convolutions
[LIST]
[*] Dirac Sequences
[*] The Weierstrass Theorem
[/LIST]
[*] Fourier Series
[LIST]
[*] Hermitian Products and Orthogonality
[*] Trigonometric Polynomials as a Total Family
[*] Explicit Uniform Approximation
[*] Pointwise Convergence
[/LIST]
[*] Improper Integrals
[LIST]
[*] Definition
[*] Criteria for Convergence
[*] Interchanging Derivatives and Integrals
[*] The Heat Kernel
[/LIST]
[*] The Fourier Integral
[LIST]
[*] The Schwartz Space
[*] The Fourier Inversion Formula
[*] An Example of Fourier Transform not in the Schwartz Space
[/LIST]
[/LIST]
[*] Calculus in Vector Spaces
[LIST]
[*] Functions on n-Space
[LIST]
[*] Partial Derivatives
[*] Differentiability and the Chain Rule
[*] Potential Functions
[*] Curve Integrals
[*] Taylor's Formula
[*] Maxima and the Derivative
[/LIST]
[*] The Winding Number and Global Potential Functions
[LIST]
[*] Another Description of the Integral Along a Path
[*] The Winding Number and Homology
[*] Proof of the Global Integrability Theorem
[*] The Integral Over Continuous Paths
[*] The Homotopy Form of the Integrability Theorem
[*] More on Homotopies
[/LIST]
[*] Derivatives in Vector Spaces
[LIST]
[*] The Space of Continuous Linear Maps
[*] The Derivative as a Linear Map
[*] Properties of the Derivative
[*] Mean Value Theorem
[*] The Second Derivative
[*] Higher Derivatives and Taylor's Formula
[*] Partial Derivatives
[*] Differentiating Under the Integral Sign
[/LIST]
[*] Inverse Mapping Theorem
[LIST]
[*] The Shrinking Lemma
[*] Inverse Mappings, Linear Case
[*] The Inverse Mapping Theorem
[*] Implicit Functions and Charts
[*] Product Decompositions
[/LIST]
[*] Ordinary Differential Equations
[LIST]
[*] Local Existence and Uniqueness
[*] Approximate Solutions
[*] Linear Differential Equations
[*] Dependence on Initial Conditions
[/LIST]
[/LIST]
[*] Multiple Integration
[LIST]
[*] Multiple Integrals
[LIST]
[*] Elementary Multiple Integration
[*] Criteria for Admissibility
[*] Repeated Integrals
[*] Change of Variables
[*] Vector Fields on Spheres
[/LIST]
[*] Differential Forms
[LIST]
[*] Definitions
[*] Stokes' Theorem for a Rectangle
[*] Inverse Image of a Form
[*] Stokes' Formula for Simplices
[/LIST]
[/LIST]
[*] Appendix
[*] Index
[/LIST]
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