Definition:Euclidean Space

Let $$\reals^n$$ be an $n$-dimensional real vector space.

Let $$M = \left\{{\reals^n, d}\right\}$$ where $$d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$$.

Then $$M = \left\{{\reals^n, d}\right\}$$ is a metric space.

Such a space is called a Euclidean $$n$$-space.

The metric $$d$$ is called the Euclidean metric.

The topology induced by the Euclidean metric on $$M = \left\{{\reals^n, d}\right\}$$ is called the Euclidean topology.

Any vector space for which a metric is defined that is precisely equivalent for any two points to the Euclidean metric is called a Euclidean space.