Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
6 - Metric Spaces
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
Summary
Some of the concepts with which we have been concerned are not special to the real or complex numbers but are connected to the general idea of distance, in particular the notion of objects being very close to each other. These ideas can be made general and precise. The abstract concept allows us to develop once and for all a vocabulary and basic results that apply in many circumstances beyond the real numbers. In particular, there are various useful senses in which functions might be considered as being very close to each other (or not).
Metrics
The abstract setting for the notion of distance is a set S, whose elements may be referred to as points. A distance function or metric on S is a function that assigns to each pair of points p and q in S a real number d(p, q) and has the properties, for all p, q, and r in S:
D1 (Positivity) d(p, q) ≥ 0; d(p, q) = 0 if and only if p = q.
D2 (Symmetry) d(p, q) = d(q, p).
D3 (Triangle inequality) d(p, r) ≤ d(p, q) + d(q, r).
Definition. A metric space is a pair (S, d), where d is a metric defined on the set S.
Examples
If S is an arbitrary set, the discrete metric on S is defined by d(p, p) = 0, while d(p, q) = 1 if q ≠ p.
Suppose that S consists of all strings of length n of 0's and 1's; two such strings of length 5 are 00101 and 10110. Let d(p, q) be the number of places in which the strings differ; for our example, d(p, q) = 3. (If q is supposed to be a copy of p, d(p, q) is the number of errors.)
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- AnalysisAn Introduction, pp. 73 - 85Publisher: Cambridge University PressPrint publication year: 2004