The Continuum

A Constructive Approach to Basic Concepts of Real Analysis
2005
ISBN: 978-3-322-82038-9
Verlag: Vieweg+Teubner Verlag

Buch, Englisch, 136 Seiten, Paperback, Format (B × H): 170 mm x 240 mm, Gewicht: 271 g

Das Buch ist gut geeignet als Textgrundlage für Seminare aus reeller Analysis (vorzugsweise für Seminare, nachdem die Studierenden den üblichen formal-axiomatischen Zugang zu den reellen Zahlen erfahren haben).


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.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Taschner, Rudolf

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik

Inhaltsverzeichnis

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.- 2.2 Order relations.- 2.3 Equality and apartness.- 2.4 Convergent sequences of real numbers.- 3 Metric spaces.- 3.1 Metric spaces and complete metric spaces.- 3.2 Compact metric spaces.- 3.3 Topological concepts.- 3.4 The s-dimensional continuum.- 4 Continuous functions.- 4.1 Pointwise continuity.- 4.2 Uniform continuity.- 4.3 Elementary calculations in the continuum.- 4.4 Sequences and sets of continuous functions.- 5 Literature.

Über Autor(innen)

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.