Table of Contents: Preface pp.i-x
Chapter 1 - Preliminaries; pp. 1-16
1.1 Sets, Relations, Functions, Cardinals and Ordinals
1.2 Reals, Some Basic Theorems and Sequence Limits
Problems
Chapter 2 - Riemann Integrals; pp. 17-44
2.1 Definitions, Examples, and Basic Properties
2.2 Algebraic Operations and the Darboux Criterion
2.3 Fundamental Theorem of Calculus
2.4 Improper Integrals
Problems
Chapter 3 - Riemann-Stieltjes Integrals; pp. 45-68
3.1 Functions of Bounded Variation
3.2 Definition and Basic Properties
3.3 Nonexistence and Existence for Integrals
3.4 Evaluations of Integrals
3.5 Improper Cases
Problems
Chapter 4 - Lebesgue-Radon-Stieltjes Integrals; pp. 69-108
4.1 Foundational Material
4.2 Essential Properties
4.3 Convergence Theorems
4.4 Extension via Measurability
4.5 Double and Iterated Integrals with Applications
Problems
Chapter 5 - Metric Spaces; pp. 109-128
5.1 Metrizable Topology
5.2 Completeness
5.3 Compactness, Density and Separability
Problems
Chapter 6 - Continuous Maps; pp. 129-148
6.1 Criteria for Continuity
6.2 Continuous Maps on Compact or Connected Spaces
6.3 Sequences of Mappings
6.4 Contractions
6.5 Equivalence of Metric Spaces
Problems
Chapter 7 - Normed Linear Spaces; pp. 149-172
7.1 Linear Spaces, Norms and Quotient Spaces
7.2 Finite Dimensional Spaces
7.3 Bounded Linear Operators
7.4 Linear Functionals via Hahn-Banach Extension
Problems
Chapter 8 - Banach Spaces via Operators and Functionals; pp. 173-204
8.1 Definition and Beginning Examples
8.2 Uniform Boundedness, Open Map and Closed Graph
8.3 Dual Banach Spaces by Examples
8.4 Weak and Weak Topologies
8.5 Compact and Dual Operators
Problems
Chapter 9 - Hilbert Spaces and Their Operators; pp. 205-238
9.1 Definition, Examples and Basic Properties
9.2 Orthogonality, Orthogonal Complement and Duality
9.3 Orthonormal Sets and Bases
9.4 Five Special Bounded Operators
9.5 Compact Operators via the Spectrum
Problems
Hints and Solutions pp.239-280
References pp.281-284
Index pp.285-287 |