Book:A.N. Kolmogorov/Introductory Real Analysis

Subject Matter

 * Analysis
 * Metric Spaces
 * Topology
 * Linear Algebra
 * Measure Theory
 * Calculus

Contents

 * Editor's Preface


 * 1. SET THEORY
 * 1. Sets and Functions
 * 1.1 Basic definitions
 * 1.2 Operations on sets
 * 1.3 Functions and mappings. Images and preimages
 * 1.4 Decomposition of a set into classes. Equivalence relations
 * 2. Equivalence of Sets. The Power of a Set
 * 2.1 Finite and infinite sets
 * 2.2 Countable sets
 * 2.3 Equivalence of sets
 * 2.4 Uncountability of the real numbers
 * 2.5 The power of a set
 * 2.6 The Cantor-Bernstein theorem
 * 3. Ordered Sets and Ordinal Numbers
 * 3.1 Partially ordered sets
 * 3.2 Order-preserving mappings. Isomorphisms
 * 3.3 Ordered sets. Order types
 * 3.4 Ordered sums and products of ordered sets
 * 3.5 Well-ordered sets. Ordinal numbers
 * 3.6 Comparison of ordinal numbers
 * 3.7 The well-ordering theorem, the axiom of choice and equivalent assertions
 * 3.8 Transfinite induction
 * 3.9 Historical remarks
 * 4. Systems of Sets
 * 4.1 Rings of sets
 * 4.2 Semirings of sets
 * 4.3 The ring generated by a semiring
 * 4.4 Borel algebras


 * 2. METRIC SPACES
 * 5. Basic Concepts
 * 5.1 Definitions and examples
 * 5.2 Continuous mappings and homeomorphisms. Isometric spaces
 * 6. Convergence. Open and closed sets
 * 6.1 Closure of a set. Limit points
 * 6.2 Convergence and limits
 * 6.3 Dense subsets. Separable spaces
 * 6.4 Closed sets
 * 6.5 Open sets
 * 6.6 Open and closed sets on the real line
 * 7. Complete metric spaces
 * 7.1 Definitions and examples
 * 7.2 The nested sphere theorem
 * 7.3 Baire's theorem
 * 7.4 Completion of a metric space
 * 8. Contraction mappings
 * 8.1 Definition of a contraction mapping. The fixed point theorem
 * 8.2 Contraction mappings and differential equations
 * 8.3 Contraction mappings and integral equations


 * 3. TOPOLOGICAL SPACES
 * 9. Basic Concepts
 * 9.1 Definitions and examples
 * 9.2 Comparison of topologies
 * 9.3 Bases. Axioms of countability
 * 9.4 Convergent sequences in a topological space
 * 9.5 Axioms of separation
 * 9.6 Continuous mappings. Homeomorphisms
 * 9.7 Various ways of specifying topologies. Metrizability
 * 10. Compactness
 * 10.1 Compact topological spaces
 * 10.2 Continuous mappings of compact spaces
 * 10.3 Countable compactness
 * 10.4 Relatively compact subsets
 * 11. Compactness in Metric Spaces
 * 11.1 Total boundedness
 * 11.2 Compactness and total boundedness
 * 11.3 Relatively compact subsets of a metric space
 * 11.4 Arzelà's Theorem
 * 11.5 Peano's Theorem
 * 12. Real Functions on Metric and Topological Spaces
 * 12.1 Continuous and uniformly continuous functions and functionals
 * 12.2 Continuous and semicontinuous functions on compact spaces
 * 12.3 Continuous curves in metric spaces
 * 4. LINEAR SPACES
 * 13. Basic Concepts
 * 13.1 Definitions and examples
 * 13.2 Linear dependence
 * 13.3 Subspaces
 * 13.4 Factor spaces
 * 13.5 Linear functionals
 * 13.6 The null space of a functional. Hyperplanes
 * 14. Convex Sets and Functionals. The Hahn-Banach Theorem
 * 14.1 Convex sets and bodies
 * 14.2 Convex functionals
 * 14.3 The Minkowski functional
 * 14.4 The Hahn-Banach theorem
 * 14.5 Separation of convex sets in a linear space
 * 15. Normed Linear Spaces
 * 15.1 Definitions and examples
 * 15.2 Subspaces of a normed linear space
 * 16. Euclidean Spaces
 * 16.1 Scalar products. Orthogonality and bases
 * 16.2 Examples
 * 16.3 Existence of an orthogonal basis. Orthogonalization
 * 16.4 Bessel's inequality. Closed orthogonal systems
 * 16.5 Complete Euclidean spaces. The Riesz-Fischer theorem
 * 16.6 Hilbert space. The isomorphism theorem
 * 16.7 Subspaces. Orthogonal complements and direct sums
 * 16.8 Characterization of Euclidean spaces
 * 16.9 Complex Euclidean spaces
 * 17. Topological Linear Spaces
 * 17.1 Definitions and examples
 * 17.2 Historical remarks


 * 5. LINEAR FUNCTIONALS
 * 18. Continuous Linear Functionals
 * 18.1 Continuous linear functionals on a topological linear space
 * 18.2 Continuous linear functionals on a normed linear space
 * 18.3 The Hahn-Banach theorem for a normed linear space
 * 19. The Conjugate Space
 * 19.1 Definition of the conjugate space
 * 19.2 The conjugate space of a normed linear space
 * 19.3 The strong topology in the conjugate space
 * 19.4 The second conjugate space
 * 20. The Weak Topology and Weak Convergence
 * 20.1 The weak topology in a topological linear space
 * 20.2 Weak convergence
 * 20.3 The weak topology and weak convergence in a conjugate space
 * 20.4 The weak* topology
 * 21. Generalized Functions
 * 21.1 Preliminary remarks
 * 21.2 The test space and test functions. Generalized functions
 * 21.3 Operations on generalized functions
 * 21.4 Differential equations and generalized functions
 * 21.5 Further developments


 * 6. LINEAR OPERATORS
 * 22. Basic Concepts
 * 22.1 Definitions and examples
 * 22.2 Continuity and boundedness
 * 22.3 Sums and products of operators
 * 23. Inverse and Adjoint Operators
 * 23.1 The inverse operator. Invertibility
 * 23.2 The adjoint operator
 * 23.3 The adjoint operator in Hilbert space. Self-adjoint operators
 * 23.4 The spectrum of an operator. The resolvent
 * 24. Completely Continuous Operators
 * 24.1 Definitions and examples
 * 24.2 Basic properties of completely continuous operators
 * 24.3 Completely continous operators in Hilbert space


 * 7. MEASURE
 * 25. Measure in the Plane
 * 25.1 Measure of elementary sets
 * 25.2 Lebesgue measure of plane sets
 * 26. General Measure Theory
 * 26.1 Measure on a semiring
 * 26.3 Countably additive measures
 * 27. Extensions of Measures


 * 8. INTEGRATION
 * 28. Measurable Functions
 * 28.1 Basic properties of measurable functions
 * 28.2 Simple functions. Algebraic operations on measurable functions
 * 28.3 Equivalent functions
 * 28.4 Convergence almost everywhere
 * 28.5 Egorov's theorem
 * 29. The Lebesgue Integral
 * 29.1 Definition and basic properties of the Lebesgue integral
 * 29.2 Some key theorems
 * 30. Further Properties of the Lebesgue Integral
 * 30.1 Passage to the limit in Lebesgue integrals
 * 30.2 The Lebesgue integral over a set of infinite measure
 * 30.3 The Lebesgue integral vs. the Riemann integral


 * 9. DIFFERENTIATION
 * 31. Differentiation of the Indefinite Lebesgue Integral
 * 31.1 Basic properties of monotonic functions
 * 31.2 Differentiation of a monotonic function
 * 31.3 Differentiation of an integral with respect to its upper limit
 * 32. Functions of a Bounded Variable
 * 33. Reconstruction of a Function from its Derivative
 * 33.1 Statement of the problem
 * 33.2 Absolutely continuous function
 * 33.3 The Lebesgue decomposition
 * 34. The Lebesgue Integral as a Set Function
 * 34.1 Charges. The Hahn and Jordan decompositions
 * 34.2 Classification of charges. The Radon-Nikodým theorem


 * 10. MORE ON INTEGRATION
 * 35. Product Measures. Fubini's Theorem
 * 35.1 Direct products of sets and measures
 * 35.2 Evaluation of a product measure
 * 35.3 Fubini's theorem
 * 36. The Stieltjes Integral
 * 36.1 Stieltjes measure
 * 36.2 The Lebesgue-Stieltjes integral
 * 36.3 Applications to probability theory
 * 36.4 The Riemann-Stieltjes integral
 * 36.5 Helly's theorems
 * 36.6 The Riesz representation theorem
 * 37. The spaces $L_1$ and $L_2$
 * 37.1 Definition and basic properties of $L_1$
 * 37.2 Definition and basic properties of $L_2$


 * Bibliography
 * Index



Source work progress
* : $\S 3.5$: Well-ordered sets. Ordinal Numbers: Example $2$