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Definition: Metric (Distance)

Let \(X\) be a set. A function mapping the Cartesian product $X\times X$ to the set of real numbers $\mathbb R$, i.e. the function \(d:X\times X\mapsto \mathbb R, (x,y)\mapsto d(x,y)\) is called a metric or a distance on \(X\), if it fulfills the following properties:

1. \(d(x,y)=0\) if and only if \(x=y\).
2. Symmetry: \(d(x,y)=d(y,x)\) for all \(x,y\in X\).
3. Symmetry: \(d(x,y)=d(y,x)\) for all \(x,y\in X\).

Table of Contents

Corollaries: 1

Mentioned in:

Corollaries: 1 2
Definitions: 3 4 5 6 7
Parts: 8 9 10
Proofs: 11 12 13 14 15 16 17
Propositions: 18 19 20 21

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984