Calculus of Real and Complex Variables

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