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 L1 and L2