The Continuum: A Constructive Approach to Basic Concepts of Real Analysis

ISBN-13: 9783322820389 / Angielski / Miękka / 2012 / 136 str.

Rudolf Taschner

Opis

"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. .

Spis treści

1 Introduction and historical remarks.- 1.1 Farey fractions.- 1.2 The pentagram.- 1.3 Continued fractions.- 1.4 Special square roots.- 1.5 Dedekind cuts.- 1.6 Weyl’s alternative.- 1.7 Brouwer’s alternative.- 1.8 Integration in traditional and in intuitionistic framework.- 1.9 The wager.- 1.10 How to read the following pages.- 2 Real numbers.- 2.1 Definition of real numbers.- 2.1.1 Decimal numbers.- 2.1.2 Rounding of decimal numbers.- 2.1.3 Definition and examples of real numbers.- 2.1.4 Differences and absolute differences.- 2.2 Order relations.- 2.2.1 Definitions and criteria.- 2.2.2 Properties of the order relations.- 2.2.3 Order relations and differences.- 2.2.4 Order relations and absolute differences.- 2.2.5 Triangle inequalities.- 2.2.6 Interpolation and Dichotomy.- 2.3 Equality and apartness.- 2.3.1 Definition and criteria.- 2.3.2 Properties of equality and apartness.- 2.4 Convergent sequences of real numbers.- 2.4.1 The limit of convergent sequences.- 2.4.2 Limit and order.- 2.4.3 Limit and differences.- 2.4.4 The convergence criterion.- 3 Metric spaces.- 3.1 Metric spaces and complete metric spaces.- 3.1.1 Definition of metric spaces.- 3.1.2 Fundamental sequences.- 3.1.3 Limit points.- 3.1.4 Apartness and equality of limit points.- 3.1.5 Sequences in metric spaces.- 3.1.6 Complete metric spaces.- 3.1.7 Rounded and sufficient approximations.- 3.2 Compact metric spaces.- 3.2.1 Bounded and totally bounded sequences.- 3.2.2 Located sequences.- 3.2.3 The infimum.- 3.2.4 The hypothesis of Dedekind and Cantor.- 3.2.5 Bounded, totally bounded, and located sets.- 3.2.6 Separable and compact spaces.- 3.2.7 Bars.- 3.2.8 Bars and compact spaces.- 3.3 Topological concepts.- 3.3.1 The cover of a set.- 3.3.2 The distance between a point and a set.- 3.3.3 The neighborhood of a point.- 3.3.4 Dense and nowhere dense.- 3.3.5 Connectedness.- 3.4 The s-dimensional continuum.- 3.4.1 Metrics in the s-dimensional space.- 3.4.2 The completion of the s-dimensional space.- 3.4.3 Cells, rays, and linear subspaces.- 3.4.4 Totally bounded sets in the s-dimensional continuum.- 3.4.5 The supremum and the infimum.- 3.4.6 Compact intervals.- 4 Continuous functions.- 4.1 Pointwise continuity.- 4.1.1 The concept of function.- 4.1.2 The continuity of a function at a point.- 4.1.3 Three properties of continuity.- 4.1.4 Continuity at inner points.- 4.2 Uniform continuity.- 4.2.1 Pointwise and uniform continuity.- 4.2.2 Uniform continuity and totally boundness.- 4.2.3 Uniform continuity and connectedness.- 4.2.4 Uniform continuity on compact spaces.- 4.3 Elementary calculations in the continuum.- 4.3.1 Continuity of addition and multiplication.- 4.3.2 Continuity of the absolute value.- 4.3.3 Continuity of division.- 4.3.4 Inverse functions.- 4.4 Sequences and sets of continuous functions.- 4.4.1 Pointwise and uniform convergence.- 4.4.2 Sequences of functions defined on compact spaces.- 4.4.3 Spaces of functions defined on compact spaces.- 4.4.4 Compact spaces of functions.- 5 Literature.

O autorze

Rudolf Taschner is Professor of Mathematics at the "Institute for Analysis and Scientific Computing", Technical University Vienna, Austria. In his recent book "Der Zahlen gigantische Schatten" (Vieweg 2004) he describes how intensively numbers penetrate the aspects of our life, and how far the "shadows of numbers" reach.

Blurb

In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the famous theorems of Brouwer concerning the continuity of all functions that are defined on "whole" intervals, the uniform continuity of all functions that are defined on compact intervals, and the uniform convergence of all pointwise converging sequences of functions defined on compact intervals. The constructive approach is interesting both in itself and as a contrast to, for example, the formal axiomatic one.