This book assumes a standard first course in analysis and gives a unified treatment
of several topics taught in the last two years of an honours degree course.
In the first chapter the standard topological notions are introduced in a metric space setting;
complete metric spaces are defined and the contraction mapping theorem and Baire's theorem proved.
The second chapter gives the standard properties of continuous
functions between metric spaces; pointwise convergence of functions is studied
and its shortcomings motivate a discussion of uniform convergence; among the
theorems proved are Dini's, the Stone-Weierstrass and the Ascoli-Arzela.
Chapter 3 gives further results on uniform convergence, in particular its relationship to the
preservation of Riemann integrability and differentiability; the former, establishing
the need for a more general integral, leads naturally to Chapter 4. Here the Daniel!
extension procedure is applied to the lattice of continuous functions of compact support
and the Riemann integral; the use of Dini's theorem makes this particularly neat.
The relationship to the measure-theoretic approach is clearly explained.
Finally double integrals are introduced and the Fubini and Tonelli theorems proved; here
the extension procedure is applied to a lattice of step functions to avoid assuming
knowledge of Riemann integration on the plane; in the reviewer's opinion it would have
been preferable to take the continuous functions of compact support and the
repeated integral, obtaining the equality of the two repeated integrals as an application
of the Stone-Weierstrass theorem. The last chapter gives the L¹ and L² theories of
the Fourier transform.
This book is very attractive: the treatment is leisurely and while it is consistently
rigorous many of the harder theorems have their proofs supplemented by informal
explanations and graphical illustrations; also a number of interesting applications
are given. Each chapter has a large number of interesting exercises some of which
extend the theory while others give relevant counter-examples. Although the reluctance
of undergraduates to buy books is to be deplored it must be reckoned with. The low price
of this book, its suitability for several courses and the high standard of
its presentation combine to make it ideal as a course book.
FREDA E. ALEXANDER
Author(s): Gordon H. Fullerton
Series: University Mathematical Texts
Publisher: Oliver and Boyd Edinburgh
Year: 1971
Language: English
Pages: 158
City: Nottingham
Tags: Mathematical Analysis, Lebesgue integration, Metric Spaces
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
