Metric Function

up:: Metric Space

A function over some set is a metric if it satisfies the following properties:

1. Non-degeneracy:
2. Triangle Inequality:
3. Positivity:
4. Commutativity:

Sufficient conditions for a metric

One can shrink these conditions to two:

1. Non-degeneracy (as above)
2. (Modified) Triangular Inequality:

From those two, one can prove the other conditions:

• :
• : and

Examples of metrics

Euclidean spaces

The initial motivation for metric spaces comes from with the Euclidean metric

Trivial metric

An example of a metric borne out of this general definition is the trivial metric:

Supremum norm and distances in function space

Consider the set of all continuous functions over , and define the supremum norm as

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 24