Rudolf Taschner. The Continuum

Transcription

1 Rudolf Taschner The Continuum

2 Rudolf Taschner The Continuum A Constructive Approach to Basic Concepts of Real Analysis ai vleweg

3 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the Internet at < Prof. Dr. Rudolf Taschner Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstr. 8 A-I040 WIEN rudolf.taschner@tuwien.ac.at First edition, September 2005 All rights reserved Friedr. Vieweg & Sohn VerlagjGWV Fachverlage GmbH, Wiesbaden 2005 Softcover reprint of the hardcover 1 st edition 2005 Editorial office: Ulrike Schmickler-Hirzebruch / Petra RuBkamp Vieweg is a company in the specialist publishing group Springer Science+Business Media. No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Cover design: Ulrike Weigel, Printed on acid-free paper ISBN-I3: e-isbn-13: DOl: /

4 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

5 Contents 1 Introduction and historical remarks F AREY fractions The pentagram Continued fractions Special square roots DEDEKIND cuts WEYL'S alternative BROUWER's alternative Integration in traditional and in intuitionistic framework The wager How to read the following pages 19 2 Real numbers Definition of real numbers Decimal numbers Rounding of decimal numbers Definition and examples of real numbers Differences and absolute differences Order relations Definitions and criteria Properties of the order relations Order relations and differences Order relations and absolute differences Triangle inequalities... 33

6 iv Contents Interpolation and Dichotomy. 2.3 Equality and apartness Definition and criteria Properties of equality and apartness 2.4 Convergent sequences of real numbers The limit of convergent sequences Limit and order Limit and differences The convergence criterion 3 Metric spaces 3.1 Metric spaces and complete metric spaces Definition of metric spaces Fundamental sequences Limit points Apartness and equality of limit points Sequences in metric spaces Complete metric spaces Rounded and sufficient approximations 3.2 Compact metric spaces Bounded and totally bounded sequences Located sequences The infimum The hypothesis of DE DE KIND and CANTOR Bounded, totally bounded, and located sets Separable and compact spaces Bars Bars and compact spaces 3.3 Topological concepts The cover of a set The distance between a point and a set The neighborhood of a point Dense and nowhere dense Connectedness The s-dimensional continuum Metrics in the s-dimensional space The completion of the s-dimensional space Cells, rays, and linear subspaces Totally bounded sets in the s-dimensional continuum The supremum and the infimum Compact intervals Continuous functions 4.1 Pointwise continuity The concept of function

7 Contents v The continuity of a function at a point Three properties of continuity Continuity at inner points Uniform continuity Pointwise and uniform continuity Uniform continuity and totally boundness Uniform continuity and connectedness Uniform continuity on compact spaces Elementary calculations in the continuum Continuity of addition and multiplication Continuity of the absolute value Continuity of division Inverse functions Sequences and sets of continuous functions Pointwise and uniform convergence Sequences of functions defined on compact spaces Spaces of functions defined on compact spaces Compact spaces of functions Literature 129 Index 134

8 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)

9 Hermann Weyl ( Peter Bettschart, Wien) LEI Brouwer ( E van Moorkorken 1943) Hermann Weyl ( Peter Bettschart, Wien)