Preface
1. Set Theory and General Topology
1.1 Basic Definitions
1.2 The Schroder Bernstein Theorem
1.3 Equivalence Relations
1.4 The Hausdorff Maximal theorem
1.5 Exercises
1.6 General Topology
1.7 Tychonoff’s theorem
1.8 Urysohn’s lemma
1.9 Exercises
2. Compactness, Continuous Functions
2.1 Compactness in Metric Space
2.2 Compactness in Spaces of Continuous Functions
2.3 Connectedness in Normed Linear Space
2.4 Stone Weierstrass theorem
2.5 An Approach to the Integral
2.6 Real Valued Functions of Many Variables
2.7 Tietze Extension Theorem
2.8 Exercises
3. Banach Spaces
3.1 Baire category Theorem
3.2 Fundamental Theorems
3.3 Hahn Banach Theorem
3.4 Exercises
4. Hilbert Spaces
4.1 Basic Theory
4.2 Finite Dimensional Normed Linear Space
4.3 Uniformly Convex Banach spaces
4.4 Hilbert Schmidt Theorem
4.5 Covering Theorems
4.6 Exercises
5. Calculus in Banach Space
5.1 The Derivative
5.2 The Mean Value Theorem
5.3 Finite Dimensions
5.4 Higher Order Derivatives .
5.5 Implicit and Inverse Function Theorems
5.6 Ordinary Differential Equations
5.7 The Brouwer Fixed Point theorem
5.8 Exercises
6. Topological Vector Spaces
6.1 Separation Theorems
6.2 The Weak and Weak* Topologies
6.3 The Tychonoff Fixed Point Theorem∗ Topologies
6.4 Set-Valued Maps
6.5 Finite Dimensional Spaces
6.6 Exercises
7. Measures and Measurable Functions
7.1 σ Algebras
7.2 Approximation with Simple Functions
7.3 Dynkin’s Lemma
7.4 Signed Measures
7.5 Exercises
8. The Abstract Lebesgue Integral
8.1 The Riemann Integral
8.2 The Lebesgue Integral for Nonnegative Functions
8.3 The Monotone Convergence Theorem
8.4 Other Definitions
8.5 The Space L1, Righteous Functionals
8.6 The Radon Nikodym Theorem
8.7 Double Sums of Nonnegative Terms
8.8 The Individual Ergodic Theorem
8.9 Exercises
9. The Construction of Measures
9.1 Outer Measures
9.2 Partition of Unity
9.3 Measures on Hausdorff Spaces
9.4 Measures and Positive Linear Functionals
9.5 Measurable Sets and Borel Sets
9.6 Some Examples
9.7 The Distribution Function
9.8 Fubini’s Theorem for Lebesgue Measure
9.9 Exercises
10. Properties of Lebesgue Measure
10.1 Translation Invariance
10.2 Maximal Functions
10.3 Vitali Coverings
10.4 Linear Change of Variables
10.5 Nonlinear Change of Variables
10.6 Spherical Coordinates
10.7 Exercises
11. Measures on Products
11.1 Product Measurability
11.2 Slicing Measures
11.3 Measures on an Infinite Product
11.4 Exercises
12. The Lp Spaces
12.1 Basic Inequalities and Properties
12.2 Density of Continuous Functions
12.3 Continuity of Translation
12.4 Separability
12.5 Mollifiers and Density of Smooth Functions
12.6 Smooth Partitions of Unity
12.7 Exercises
13. Representation Theorems
13.1 Radon Nikodym Theorem
13.2 Vector Measures
13.3 Representation for the Dual Space of Lp
13.4 Weak Compactness
13.5 The Dual Space of L∞(Ω)
13.6 The Dual Space of C0(X)
13.7 Non σ Finite Measure Spaces
13.8 Exercises
14. General Radon Measures
14.1 Vitali Coverings
14.2 Differentiation of Increasing Functions
14.3 Symmetric Derivative for Radon Measures
14.4 The Radon Nikodym theorem for Radon measures
14.5 Young Measures
14.6 Exercises
15. Fourier Transforms
15.1 Fourier Transforms of Functions in G
15.2 Fourier Transforms of Just About Anything
15.3 Weak Derivatives
15.4 Exercises
16. Fourier Analysis in Rn
16.1 The Marcinkiewicz Interpolation Theorem
16.2 The Calderon Zygmund Decomposition
16.3 Mihlin’s Theorem
16.4 Singular Integrals
16.5 The Helmholtz Decomposition of Vector Fields
16.6 Exercises
17. Probability
17.1 Random Vectors
17.2 Characteristic and Moment Generating Functions
17.3 The Continuity Theorem .
17.4 Conditional Probability and Independence
17.5 Conditional Expectation
17.6 Conditional Expectation Given a σ Algebra
17.7 Strong Law of Large Numbers
17.8 The Normal Distribution
17.9 Central Limit Theorem
17.10 Exercises
18. Hausdorff Measure
18.1 Lipschitz Functions
18.2 Lipschitz Functions and Gateaux Derivatives
18.3 Rademacher’s Theorem
18.4 Definition of Hausdorff Measures
18.5 Properties of Hausdorff Measure
18.6 H p and mp
18.7 Technical Considerations
18.8 The Proper Value of β(p)
18.9 A Formula for α(p)
18.10 Exercises
19. The Area Formula
19.1 Estimates for Hausdorff Measure
19.2 Comparison Theorems
19.3 The Area Formula
19.4 The Divergence Theorem
19.5 The Coarea Formula
19.6 Change of Variables
19.7 Exercises
20. Integration for Vector Valued Functions
20.1 Strong and Weak Measurability
20.2 Eggoroff’s Theorem
20.3 The Bochner Integral
20.4 The Spaces Lp(Ω;X)
20.5 Measurable Representatives
20.6 Vector Measures
20.7 The Riesz Representation Theorem
20.8 Exercises
21. Convex Functions
21.1 Continuity Properties
21.2 Separation Properties
21.3 Conjugate Functions
21.4 Subgradients
21.5 Subgradients in Hilbert Space
21.6 Exercises
A. Multifunctions and Their Measurability
A.1 The General Case
A.2 A Special Case When Γ(ω) Compact
A.3 Kuratowski’s Theorem
A.4 Measurability of Fixed Points
A.5 Other Measurability Considerations
B. Stone’s Theorem and Partitions of Unity
B.1 An Extension Theorem
C. Orlitz Spaces
C.1 Dual Spaces in Orlitz Space
D. Analytic Functions
D.1 Ordinary Differential Equations
Biography
Kenneth Kuttler is an emeritus professor at Brigham Young University, who holds his PhD from University of Texas. His primary area of research is Partial Differential Equations and Inclusions.
