Calculus of Real and Complex Variables [January 11, 2021 ed.]

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English Pages 544 Year 2021

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Kenneth Kuttler

Categories
Mathematics
Analysis

Table of contents :
I Preliminary Topics
Basic Notions
Sets and Set Notation
The Schroder Bernstein Theorem
Equivalence Relations
The Hausdorff Maximal Theorem
The Hamel Basis
Analysis of Real and Complex Numbers
Roots Of Complex Numbers
The Complex Exponential
The Cauchy Schwarz Inequality
Polynomials and Algebra
The Fundamental Theorem of Algebra
Some Topics from Analysis
lim sup and lim inf
Nested Interval Lemma
Exercises
Basic Topology and Algebra
Some Algebra
Metric Spaces
Closed and Open Sets
Sequences and Cauchy Sequences
Separability and Complete Separability
Compactness
Continuous Functions
Limits of Vector Valued Functions
The Extreme Value Theorem and Uniform Continuity
Convergence of Functions
Multiplication of Series
Tietze Extension Theorem
Root Test
Equivalence of Norms
Norms on Linear Maps
Connected Sets
Stone Weierstrass Approximation Theorem
The Bernstein Polynomials
The Case of Compact Sets
The Case of a Closed Set in Rp
The Case Of Complex Valued Functions
Brouwer Fixed Point Theorem
Simplices and Triangulations
Labeling Vertices
The Brouwer Fixed Point Theorem
Exercises
II Real Analysis
The Derivative, a Linear Transformation
Basic Definitions
The Chain Rule
The Matrix of the Derivative
Existence of the Derivative, C1 Functions
Mixed Partial Derivatives
A Cofactor Identity
Implicit Function Theorem
More Continuous Partial Derivatives
Invariance of Domain
Exercises
Line Integrals
Existence and Definition
Change of Parameter
Existence
The Riemann Integral
Estimates and Approximations
Finding the Length of a C1 Curve
Curves Defined in Pieces
A Physical Application, Work
Conservative Vector Fields
Orientation
Exercises
Measures And Measurable Functions
Measurable Functions
Measures and Their Properties
Dynkin's Lemma
Measures and Outer Measures
An Outer Measure on P( R)
Measures from Outer Measures
When is a Measure a Borel Measure?
One Dimensional Lebesgue Measure
Exercises
The Abstract Lebesgue Integral
Definition For Nonnegative Measurable Functions
Riemann Integrals For Decreasing Functions
The Lebesgue Integral For Nonnegative Functions
The Lebesgue Integral for Nonnegative Simple Functions
The Monotone Convergence Theorem
Other Definitions
Fatou's Lemma
The Integral's Righteous Algebraic Desires
The Lebesgue Integral, L1
The Dominated Convergence Theorem
Product Measures
Some Important General Theorems
Eggoroff's Theorem
The Vitali Convergence Theorem
Radon Nikodym Theorem
Exercises
Positive Linear Functionals
Partitions of Unity
Positive Linear Functionals and Measures
Lebesgue Measure
Computation with Iterated Integrals
Approximation with G0=x"010E and F0=x"011B Sets and Translation Invariance
The Vitali Covering Theorems
Exercises
Basic Function Spaces
Bounded Continuous Functions
Compactness in C( K,Rn)
The Lp Spaces
Approximation Theorems
Maximal Functions and Fundamental Theorem of Calculus
A Useful Inequality
Exercises
Change of Variables
Lebesgue Measure and Linear Transformations
Change of Variables Nonlinear Maps
Mappings Which are Not One to One
Spherical Coordinates In p Dimensions
Approximation with Smooth Functions
Continuity Of Translation
Separability
Exercises
Some Fundamental Functions and Transforms
Gamma Function
Laplace Transform
Fourier Transform
Fourier Transforms in Rn
Fourier Transforms Of Just About Anything
Fourier Transforms in G
Fourier Transforms of Functions In L1( Rn)
Fourier Transforms of Functions In L2( Rn)
The Schwartz Class
Convolution
Exercises
Degree Theory, an Introduction
Sard's Lemma and Approximation
Properties of the Degree
Borsuk's Theorem
Applications
Product Formula, Jordan Separation Theorem
The Jordan Separation Theorem
Exercises
Green's Theorem
An Elementary Form of Green's Theorem
Stoke's Theorem
A General Green's Theorem
The Jordan Curve Theorem
Green's Theorem for a Rectifiable Jordan Curve
Orientation of a Jordan Curve
III Abstract Analysis
Banach Spaces
Theorems Based On Baire Category
Baire Category Theorem
Uniform Boundedness Theorem
Open Mapping Theorem
Closed Graph Theorem
Basic Theory of Hilbert Spaces
Hahn Banach Theorem
Partially Ordered Sets
Gauge Functions And Hahn Banach Theorem
The Complex Version Of The Hahn Banach Theorem
The Dual Space And Adjoint Operators
Exercises
Representation Theorems
Radon Nikodym Theorem
Vector Measures
The Dual Space of Lp( )
The Dual Space Of L( )
The Dual Space Of C0( Rp)
Exercises
IV Complex Analysis
Fundamentals
Banach Spaces
The Cauchy Riemann Equations
Contour Integrals
Primitives and Cauchy Goursat Theorem
Functions Differentiable on a Disk, Zeros
The General Cauchy Integral Formula
Riemann sphere
Exercises
Isolated Singularities and Analytic Functions
Open Mapping Theorem for Complex Valued Functions
Functions Analytic on an Annulus
The Complex Exponential and Winding Number
Cauchy Integral Formula for a Cycle
An Example of a Cycle
Isolated Singularities
The Residue Theorem
Evaluation of Improper Integrals
The Inversion of Laplace Transforms
Exercises
Mapping Theorems
Meromorphic Functions
Meromorphic on Extended Complex Plane
Rouche's Theorem
Fractional Linear Transformations
Some Examples
Riemann Mapping Theorem
Montel's Theorem
Regions with Square Root Property
Exercises
Spectral Theory of Linear Maps *
The Resolvent and Spectral Radius
Functions of Linear Transformations
Invariant Subspaces
Review of Linear Algebra
Systems of Equations
Matrices
Subspaces and Spans
Application to Matrices
Mathematical Theory of Determinants
The Function sgn
Determinants
Definition of Determinants
Permuting Rows or Columns
A Symmetric Definition
Alternating Property of the Determinant
Linear Combinations and Determinants
Determinant of a Product
Cofactor Expansions
Formula for the Inverse
Cramer's Rule
Upper Triangular Matrices
Cayley-Hamilton Theorem
Eigenvalues and Eigenvectors of a Matrix
Definition of Eigenvectors and Eigenvalues
Triangular Matrices
Defective and Nondefective Matrices
Diagonalization
Schur's Theorem
Hermitian Matrices
Right Polar Factorization
Direct Sums
Block Diagonal Matrices