Mathematical Analysis 0050023462, 9780050023464

Table of contents :
CONTENTS

CHAPTER 1 METRIC SPACES
1 Metric and normed spaces.............. 1
2 Open and closed sets.............. 4
3 Compactness.............. 1O
4 Connectedness.............. 17
5 Convergence.............. 22
6 Consequences of completeness.............. 26
Problems 1.............. 28

CHAPTER 2 CONTINUOUS FUNCTIONS
7 Definition and topological conditions.............. 32
8 Preservation of compactness and connectedness.............. 34
9 Uniform convergence.............. 39
10 Uniform continuity.............. 44
11 Weierstrass‘s Theorem.............. 46
12 The Stone—Weierstrass Theorem.............. 50
13 Compactness in C(X).............. 55
14 Topological spaces: an aside.............. 58
Problems 2.............. 59

CHAPTER 3 FURTHER RESULTS ON UNIFORM CONVERGENCE
15 Uniform convergence and integration.............. 63
16 Uniform convergence and differentiation.............. 68
17 Uniform convergence of series.............. 70
18 Tests for uniform convergence of series.............. 71
19 Power series.............. 75
Problems 3.............. 80

CHAPTER 4 LEBESGUE INTEGRATION
20 The collection K and null sets.............. 85
21 The Lebesgue integral.............. 91
22 Convergence theorems.............. 97
23 Relation between Riemann and Lebesgue integration........... 102
24 Daniell integrals.............. 109
25 Measurable functions and sets.............. 113
26 COmplex-valued functions: Lᵖ spaces.............. 120
27 Double integrals.............. 127
Problems 4.............. 133

CHAPTER 5 FOURIER TRANSFORMS
28 L¹ theory: elementary results.............. 138
29 The inversion theorem.............. 140
30 L² theory.............. 144
Problems 5.............. 148


Index 150