In this post, we will see the book What Is Distance? by Yu. A. Shreider. This is part of Popular Lectures in Mathematics series.

About the book

This introduction to the theory of metric spaces carries the student through the motivation, development, and application of an abstract mathematical concept. Yu. A. Shreider begins with a definition of mathematical concepts and then, to motivate his definition of “ metric space,” the author evaluates distance functions and their properties in elementary geometry. After introducing two sets of metric space axioms, he surveys the properties of open balls, sequences, Cauchy sequences, complete metric spaces, and isometries. In the following chapter, the author offers a number of informative examples.

Professor Shreider turns next to applications of the theories, especially the theory of coding and cybernetics. Then returning to theory, the author defines an n-dimensional vector space over real numbers and develops the theory of norms and metrics
on Euclidean n-space. The final chapter considers the possibility of generalizing the concept of metric space.

The book was adapted from the Russian edition by Leslie Cohn and Harvey Edelberg
and was published in 1974.

Credits to the original uploader.

You can get the book here.

PS: This is the 500th post on the blog! Thanks for all the support over the years!

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Contents

Introduction vii

1. The Definition of Mathematical Concepts 1

2. Distance and Its Properties in Elementary Geometry 5

3. The Definition of a Metric Space and of Distance 11

4. Some Examples of Metric Spaces 18

5. The Space of Information 27

6. Automatic Correction of Errors in Messages 38

7. Metrics and Norms in Multi-dimensional Spaces 47

8. The Smoothing of Errors in Experimental Measurements 60

9. A More General Definition of Distance 65