Preface
1 Prelude to Modern Analysis
1.1 Introduction
1.2 Sets and numbers
1.3 Functions or mappings
1.4 Countability
1.5 Point sets
1.6 Open and closed sets
1.7 Sequences
1.8 Series
1.9 Functions of a real variable
1.10 Uniform convergence
1.11 Some linear algebra
1.12 Setting off
2 Metric Spaces
2.1 Definition of a metric space
2.2 Examples of metric spaces
2.3 Solved problems
2.4 Exercises
2.5 Convergence in a metric space
2.6 Examples on pleteness
2.7 Subspace of a metric space
2.8 Solved problems
2.9 Exercises
The Fixed Point Theorem and its Applications
3.1 Mappings between metric spaces
3.2 The fixed point theorem
3.3 Applications
3.4 Perturbation mappings
3.5 Exercises
4 Compactness
4.1 Compact sets
4.2 Ascoli's theorem
4.3 Application to appromation theory
4.4 Solved problems
4.5 Exercises
5 Topological Spaces
5.1 Definitions and examples
5.2 Closed sets
5.3 Compact sets
5.4 Continuity in topological spaces
5.5 Homeomorphisms; connectedness
5.6 Solved problems
5.7 Exercises
6 Normed Vector Spaces
6.1 Definition of a normed vector space; examples
6.2 Convergence in normed spaces
6.3 Solved problems
6.4 Exercises
6.5 Finite-dimensional normed vector spaces