Definition:Euclidean Metric

Definition

Consider the Euclidean space $\left({\R^n, d}\right)$.

The metric $d$ on such a space, defined as:

$\displaystyle d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

is called the Euclidean metric.

This is sometimes also referred to as the usual metric.

Real Number Line

On the real number line, it can be seen that this definition degenerates to:

$d \left({x, y}\right) = \sqrt {\left({x - y}\right)^2} = \left|{x - y}\right|$

See absolute value.

See also

It can be proved that the Euclidean Metric is a Metric.

From Metric Induces a Topology, it follows that the Euclidean space is also a topological space.

In this context, the topology induced by the Euclidean metric is sometimes called the usual topology.

• Results about Euclidean spaces can be found here.

Source of Name

This entry was named for Euclid.

Bear in mind that Euclid himself did not in fact conceive of the Euclidean metric. It is called that because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28: \ 9$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Example $2.2.1$