## Introductory Real Analysis

By Kolmogorov and Fomin.

These are my proposed solutions. If you find any mistakes, please let me know.

Low priority project, so this will likely take me a long time! :-)

**Impressions**: This is not the original text by Kolmogorov, but is a highly

edited version by the translator. The exercises seem to be made in the translation

only. Parts are a bit unclear and uses notions that have not been introduced earlier.

Makes a lot of assumptions on previous exposure, e.g. the Heine-Borel theorem on page 92

that the reader "presumably already knows". So this is not really an introduction to

to Real Analysis, and makes references to "Elementary Analysis".

#### Chapter 1 - Set Theory

1.1 Sets and Functions

1.2 Equivalence of Sets. The Power of a Set

1.3 Ordinal Sets and Ordinal Numbers [No solutions]

1.4 Systems of Sets

#### Chapter 2 - Metric Spaces

2.5 Basic Concepts

2.6 Convergence. Open and Closed Sets

2.7 Complete Metric Spaces

2.8 Contraction Mappings

#### Chapter 3 - Topological Spaces

3.9 Basic Concepts

3.10 Compactness

3.11 Compactness in Metric Spaces

3.12 Real Functions on Metric and Topological Spaces [No solutions]

#### Chapter 4 - Linear Spaces

4.13 Basic Concepts

4.14 Convex Sets and Functionals. The Hahn-Banach Theorem

4.15 Normed Linear Spaces

4.16 Euclidean Spaces

4.17 Topological Linear Spaces

#### Chapter 5 - Linear Functionals

5.18 Continuous Linear Functionals

5.19 The Conjugate Space

5.20 The Weak Topology and Weak Convergence

5.21 Generalized Functionals

#### Chapter 6 - Linear Operators

6.22 Basic Concepts

6.23 Inverse and Adjoint Operators

6.24 Completely Continuous Operators

#### Chapter 7 - Measure

7.25 Measure in the Plane

7.26 General Measure Theory

7.27 Extensions of Measures

#### Chapter 8 - Integration

8.28 Measurable Functions

8.29 The Lebesgue Integral

8.30 Further Properties of the Lebesgue Integral

#### Chapter 9 - Differentiation

9.31 Differentiation of the Indefinite Lebesgue Integral

9.32 Functions of Bounded Variation

9.33 Reconstruction of a Function from Its Derivative

9.34 The Lebesgue Integral as a Set Function

#### Chapter 10 - More on Integration

10.35 Product Measures. Fubini's Theorem

10.36 The Stieltjes Integral

10.37 The Spaces L_{1} and L_{2}