The Continuum. A constructive approach to basic concepts of real analysis

142 Pages·2005·4.977 MB·English

by  Rudolf Taschner

Rudolf Taschner
The Continuum
Rudolf Taschner
The Continuum
A Constructive Approach
to Basic Concepts of Real Analysis
aI
vleweg
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Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie;
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Prof. Dr. Rudolf Taschner
Institute for Analysis and
Scientific Computing
Vienna University of Technology
Wiedner Hauptstr. 8
A-I040 WIEN
E-Mail: [email protected]
First edition, September 2005
All rights reserved
© Friedr. Vieweg & Sohn VerlagjGWV Fachverlage GmbH, Wiesbaden 2005
Softcover reprint of the hardcover 1s t edition 2005
Editorial office: Ulrike Schmickler-Hirzebruch / Petra RuBkamp
Vieweg is a company in the specialist publishing group Springer Science+Business Media.
www.vieweg.de
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or transmitted, mechanical, photocopying or otherwise without prior
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Cover design: Ulrike Weigel, www.CorporateDesignGroup.de
Printed on acid-free paper
ISBN-I3: 978-3-322-82038-9 e-ISBN-13: 978-3-322-82036-5
DOl: 10.1 007/978-3-322-82036-5
Preface
"Few mathematical structures have undergone as many revlSlons or have been
presented in as many guises as the real numbers. Every generation re-examines the
reals in the light of its values and mathematical objectives." This citation is said to
be due to Gian-Carlo Rota, and in this book its correctness again is affirmed. Here I
propose to investigate the structure of the mathematical continuum by undertaking a
rather unconventional access to the real numbers: the intuitionistic one. The traces
can be tracked back at least to L.E.J. Brouwer and to H. Weyl. Largely unknown
photographies of Weyl in Switzerland after World War II provided by Peter
Bettschart enliven the abstract text full of subtle definitions and sophisticated
estimations.
The book can be read by students who have undertaken the usual analysis
courses and want to know more about the intrinsic details of the underlying
concepts, and it can also be used by university teachers in lectures for advanced
undergraduates and in seminaries for graduate students.
I wish to thank Walter Lummerding and Gottfried Oehl who helped me with
their impressive expert knowledge of the English language. I also take the
opportunity to express my gratitude to Ulrike Schmickler-Hirzebruch and to the staff
of Vieweg-Verlag for editing my manuscript just now, exactly 50 years after the
death of Hermann Weyl, in their renowned publishing house.
Vienna, 2005 Rudolf Taschner
Contents
1 Introduction and historical remarks 1
1.1 F AREY fractions. . 1
1.2 The pentagram ... 3
1.3 Continued fractions . 6
1.4 Special square roots . 8
1.5 DEDEKIND cuts . . . 9
1.6 WEYL'S alternative 12
1.7 BROUWER's alternative. 13
1.8 Integration in traditional and in intuitionistic framework . 15
1.9 The wager ............ 17
1.10 How to read the following pages 19
2 Real numbers 21
2.1 Definition of real numbers ...... 21
2.1.1 Decimal numbers . ...... 21
2.1.2 Rounding of decimal numbers 23
2.1.3 Definition and examples of real numbers 24
2.1.4 Differences and absolute differences . 26
2.2 Order relations ............. 27
2.2.1 Definitions and criteria . . . . . 27
2.2.2 Properties of the order relations 29
2.2.3 Order relations and differences . 31
2.2.4 Order relations and absolute differences 32
2.2.5 Triangle inequalities .......... 33
iv Contents
2.2.6 Interpolation and Dichotomy . 35
2.3 Equality and apartness ....... . 38
2.3.1 Definition and criteria ... . 38
2.3.2 Properties of equality and apartness 40
2.4 Convergent sequences of real numbers . . 41
2.4.1 The limit of convergent sequences 41
2.4.2 Limit and order . . . . . . 42
2.4.3 Limit and differences . . . 44
2.4.4 The convergence criterion 46
3 Metric spaces 49
3.1 Metric spaces and complete metric spaces 49
3.1.1 Definition of metric spaces 49
3.1.2 Fundamental sequences .... . 51
3.1.3 Limit points ........... . 54
3.1.4 Apartness and equality of limit points 57
3.1.5 Sequences in metric spaces ..... . 58
3.1.6 Complete metric spaces ...... . 60
3.1.7 Rounded and sufficient approximations 61
3.2 Compact metric spaces ............ . 64
3.2.1 Bounded and totally bounded sequences. 64
3.2.2 Located sequences ........... . 65
3.2.3 The infimum .............. . 67
3.2.4 The hypothesis of DE DE KIND and CANTOR. 70
3.2.5 Bounded, totally bounded, and located sets 71
3.2.6 Separable and compact spaces 72
3.2.7 Bars .......... . 74
3.2.8 Bars and compact spaces 76
3.3 Topological concepts ..... . 78
3.3.1 The cover of a set ... . 78
3.3.2 The distance between a point and a set. 79
3.3.3 The neighborhood of a point 80
3.3.4 Dense and nowhere dense 82
3.3.5 Connectedness ...... . 84
3.4 The s-dimensional continuum .. . 85
3.4.1 Metrics in the s-dimensional space. 85
3.4.2 The completion of the s-dimensional space 86
3.4.3 Cells, rays, and linear subspaces ..... . 89
3.4.4 Totally bounded sets in the s-dimensional continuum 90
3.4.5 The supremum and the infimum 90
3.4.6 Compact intervals ...... . 92
4 Continuous functions 95
4.1 Pointwise continuity ..... . 95
4.1.1 The concept of function 95
Contents v
4.1.2 The continuity of a function at a point 96
4.1.3 Three properties of continuity 98
4.1.4 Continuity at inner points .... 102
4.2 Uniform continuity . . . . . . . . . . . . 105
4.2.1 Pointwise and uniform continuity 105
4.2.2 Uniform continuity and totally boundness 107
4.2.3 Uniform continuity and connectedness. 107
4.2.4 Uniform continuity on compact spaces. . 109
4.3 Elementary calculations in the continuum . . . . 110
4.3.1 Continuity of addition and multiplication 110
4.3.2 Continuity of the absolute value 111
4.3.3 Continuity of division ....... 113
4.3.4 Inverse functions . . . . . . . . . . 115
4.4 Sequences and sets of continuous functions 118
4.4.1 Pointwise and uniform convergence 118
4.4.2 Sequences of functions defined on compact spaces 121
4.4.3 Spaces of functions defined on compact spaces 122
4.4.4 Compact spaces of functions . . . . . . . . . . . . 124
5 Literature 129
Index 134
Hermann Hesse the Author of "The Glass Bead Game" and Hermann Weyl
(© Peter Bettschart, Wien)
Hermann WeyJ (© Fr Schmelhaus, Zurich) Hermann WeyJ (© Peter Bettschart, Wien)
Hermann Weyl (© Peter Bettschart, Wien) LEI Brouwer (© E van Moorkorken 1943)
Hermann Weyl (© Peter Bettschart, Wien)
1
Introduction and historical remarks
It will come as no surprise to the reader to note that the title "The Continuum"
refers to HERMANN WEYL'S renounced book on the continuum, and in fact: the
author of this book, though light-years away from the mathematical and philo
sophical capabilities of WEYL, shares his scepticism about the foundation of
analysis in the sense of GEORG CANTOR and RICHARD DEDEKIND.
In order to understand the sceptical position of WEYL, it is advisable to call
the idea of DEDEKIND cuts to our mind. We will do this by enumerating all ra
tional numbers, and will show how unattainably sharp-edged DEDEKIND cuts are
supposed to be.
1.1 FA REY fractions
In the beginning of the nineteenth century, the geologist JOHN FAREY constructed
a table of fractions in the following way: In the first row he wrote 0/1 and 1/1.
For k = 2,3, ... he used the rule: Form the k-th row by copying the (k - 1)-st
+ +
in order, but insert the fraction (p q) / (n m), the so-called mediant, between
+ :s
the consecutive fractions p / nand q / m of the (k - 1) -st row if n m k. Thus,
+ :s + +
since 1 1 2 FAREY inserted (0 1) / (1 1) between 0/1 and 1/1 and
obtained 0/1, 1/2, 1/1 for the second row. His third row was 0/1, 1/3, 1/2,2/3,
+ + + +
1/1. To obtain the fourth row he inserted (0 1) / (1 3) and (2 1) / (3 1)
+ + + +
but not (1 1) / (3 2) and (1 2) / (2 3). The first five rows of FAREY'S
R. Taschner, The Continuum
© Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden 2005