Real and Abstract Analysis

I Topology, Continuity, Algebra, Derivatives
Some Basic Topics
Basic Definitions
The Schroder Bernstein Theorem
Equivalence Relations
sup and inf
Double Series
lim sup and lim inf
Nested Interval Lemma
The Hausdorff Maximal Theorem
Metric Spaces
Open and Closed Sets, Sequences, Limit Points
Cauchy Sequences, Completeness
Closure of a Set
Separable Metric Spaces
Compact Sets
Continuous Functions
Continuity and Compactness
Lipschitz Continuity and Contraction Maps
Convergence of Functions
Compactness in C( X,Y) Ascoli Arzela Theorem
Connected Sets
Partitions of Unity in Metric Space
Exercises
Linear Spaces
Algebra in Fn, Vector Spaces
Subspaces Spans and Bases
Inner Product and Normed Linear Spaces
The Inner Product in Fn
General Inner Product Spaces
Normed Vector Spaces
The p Norms
Orthonormal Bases
Equivalence of Norms
Covering Theorems
Vitali Covering Theorem
Besicovitch Covering Theorem
Exercises
Functions on Normed Linear Spaces
L( V,W) as a Vector Space
The Norm of a Linear Map, Operator Norm
Continuous Functions in Normed Linear Space
Polynomials
Weierstrass Approximation Theorem
Functions of Many Variables
A Generalization with Tietze Extension Theorem
An Approach to the Integral
The Stone Weierstrass Approximation Theorem
Connectedness in Normed Linear Space
Saddle Points
Exercises
Fixed Point Theorems
Simplices and Triangulations
Labeling Vertices
The Brouwer Fixed Point Theorem
The Schauder Fixed Point Theorem
The Kakutani Fixed Point Theorem
Exercises
The Derivative
Limits of a Function
Basic Definitions
The Chain Rule
The Matrix of the Derivative
A Mean Value Inequality
Existence of the Derivative, C1 Functions
Higher Order Derivatives
Some Standard Notation
The Derivative and the Cartesian Product
Mixed Partial Derivatives
A Cofactor Identity
Newton's Method
Exercises
Implicit Function Theorem
Statement and Proof of the Theorem
More Derivatives
The Case of Rn
Exercises
The Method of Lagrange Multipliers
The Taylor Formula
Second Derivative Test
The Rank Theorem
The Local Structure of C1 Mappings
Invariance of Domain
Exercises
II Integration
Abstract Measures and Measurable Functions
Simple Functions and Measurable Functions
Measures and their Properties
Dynkin's Lemma
Measures and Regularity
Outer Measures
Exercises
An Outer Measure on P( R)
Measures From Outer Measures
When do the Measurable Sets Include Borel Sets?
One Dimensional Lebesgue Stieltjes Measure
Completion of a Measure Space
Vitali Coverings
Differentiation of Increasing Functions
Exercises
Multifunctions and Their Measurability
The General Case
A Special Case When ( 0=x"0121) Compact
Kuratowski's Theorem
Measurability of Fixed Points
Other Measurability Considerations
Exercises
The Abstract Lebesgue Integral
Definition for Nonnegative Measurable Functions
Riemann Integrals for Decreasing Functions
The Lebesgue Integral for Nonnegative Functions
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
Some Important General Theory
Eggoroff's Theorem
The Vitali Convergence Theorem
One Dimensional Lebesgue Stieltjes Integral
The Distribution Function
Good Lambda Inequality
Radon Nikodym Theorem
Abstract Product Measures and Integrals
Exercises
Regular Measures
Regular Measures in a Metric Space
Differentiation of Radon Measures
Maximal Functions, Fundamental Theorem of Calculus
Symmetric Derivative for Radon Measures
Radon Nikodym Theorem for Radon Measures
Absolutely Continuous Functions
Constructing Measures from Functionals
The p Dimensional Lebesgue Measure
Exercises
Change of Variables, Linear Maps
Differentiable Functions and Measurability
Change of Variables, Nonlinear Maps
Mappings which are not One to One
Spherical Coordinates
Exercises
Integration on Manifolds
Relatively Open Sets
Manifolds
The Area Measure on a Manifold
Exercises
Divergence Theorem
Volumes of Balls in Rp
Exercises
The Lp Spaces
Basic Inequalities and Properties
Density Considerations
Separability
Continuity of Translation
Mollifiers and Density of Smooth Functions
Smooth Partitions of Unity
Exercises
Degree Theory
Sard's Lemma and Approximation
Properties of the Degree
Borsuk's Theorem
Applications
Product Formula, Jordan Separation Theorem
The Jordan Separation Theorem
Uniqueness of the Degree
Exercises
Hausdorff Measure
Lipschitz Functions
Lipschitz Functions and Gateaux Derivatives
Rademacher's Theorem
Weak Derivatives
Definition of Hausdorff Measures
Properties of Hausdorff Measure
Hp and mp
Technical Considerations
Steiner Symmetrization
The Isodiametric Inequality
The Proper Value of 0=x"010C( p)
A Formula for 0=x"010B( p)
The Area Formula
Estimates for Hausdorff Measure
Comparison Theorems
A Decomposition
Estimates and a Limit
The Area Formula
Mappings that are Not One to One
The Divergence Theorem
The Reynolds Transport Formula
The Coarea Formula
Change of Variables
Orientation in Higher Dimensions
The Wedge Product
The Exterior Derivative
Stoke's Theorem
Green's Theorem and Stokes Theorem
The Divergence Theorem
Exercises
III Abstract Theory
Hausdorff Spaces And Measures
General Topological Spaces
The Alexander Sub-basis Theorem
Stone Weierstrass Theorem
The Case of Locally Compact Sets
The Case of Complex Valued Functions
Partitions of Unity
Measures on Hausdorff Spaces
Measures and Positive Linear Functionals
Slicing Measures
Exercises
Product Measures
Measure on Infinite Products
Algebras
Caratheodory Extension Theorem
Kolmogorov Extension Theorem*
Exercises
Banach Spaces
Theorems Based on Baire Category
Baire Category Theorem
Uniform Boundedness Theorem
Open Mapping Theorem
Closed Graph Theorem
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
Uniform Convexity of Lp
Closed Subspaces
Weak And Weak Topologies
Basic Definitions
Banach Alaoglu Theorem
Eberlein Smulian Theorem
Differential Equations
Exercises
Hilbert Spaces
Basic Theory
The Hilbert Space L( U)
Approximations in Hilbert Space
Orthonormal Sets
Compact Operators
Compact Operators in Hilbert Space
Nuclear Operators
Hilbert Schmidt Operators
Square Roots
Ordinary Differential Equations in Banach Space
Fractional Powers of Operators
General Theory of Continuous Semigroups
An Evolution Equation
Adjoints, Hilbert Space
Adjoints, Reflexive Banach Space
Exercises
Representation Theorems
Radon Nikodym Theorem
Vector Measures
Representation for the Dual Space of Lp
The Dual Space of L( )
Non 0=x"011B Finite Case
The Dual Space of C0( X)
Extending Righteous Functionals
The Riesz Representation Theorem
Exercises
Fourier Transforms
Fourier Transforms of Functions In G
Fourier Transforms of just about Anything
Fourier Transforms of G
Fourier Transforms of Functions in L1( Rn)
Fourier Transforms Of Functions In L2( Rn)
The Schwartz Class
Convolution
Exercises
The Bochner Integral
Strong and Weak Measurability
Eggoroff's Theorem
The Bochner Integral
Definition and Basic Properties
Taking a Closed Operator Out of the Integral
Operator Valued Functions
Review of Hilbert Schmidt Theorem
Measurable Compact Operators
Fubini's Theorem for Bochner Integrals
The Spaces Lp( ;X)
Measurable Representatives
Vector Measures
The Riesz Representation Theorem
An Example of Polish Space
Pointwise Behavior of Weakly Convergent Sequences
Some Embedding Theorems
Conditional Expectation in Banach Spaces
Exercises
Review of Some Linear Algebra
The Matrix of a Linear Map
Block Multiplication of Matrices
Schur's Theorem
Hermitian and Symmetric Matrices
The Right Polar Factorization
Elementary matrices
The Row Reduced Echelon Form Of A Matrix
Finding the Inverse of a Matrix
The Mathematical Theory of Determinants
The Function sgn
The Definition of the Determinant
A Symmetric Definition
Basic Properties of the Determinant
Expansion Using Cofactors
A Formula for the Inverse
Cramer's Rule
Rank of a Matrix
An Identity of Cauchy
The Cayley Hamilton Theorem
Stone's Theorem and Partitions of Unity
Partitions of Unity and Stone's Theorem
An Extension Theorem, Retracts