Examples of metric spaces ↩

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created: 2021-12-15 11:04:35
modified: 2022-01-10 04:13:04

Real line

is the set of all real numbers, taken with the usual metric defined by

Euclidean plane

The metric space , called the Euclidean plane is obtained if we take the set of ordered pairs of real numbers , , etc. and the Euclidean metric defined by

Another metric space is obtained if we choose the same set, but another metric defined by

This illustrates that from a given set we can obtain various metric spaces by choosing different metrics.

Function space

As a set we take the set of all real-valued functions which are functions of an independent real variable and are defined and continuous on a given closed interval . Choosing the metric defined by

where denotes the maximum, we obtain a metric space which is denoted by . This is a function space because every point of is a function.

Discrete metric space

We take any set and on it the so-called discrete metric for , defined by

This space is called a discrete metric space. (It rarely occurs in applications.)