Definition:Euclidean Space

Consider the set S whose elements are all possible n-tuples $$x=\hat{x}=(x_1,x_2,...,x_n)$$, where each of the $$x_i \in \mathbb{R}$$.

Define a metric on S by

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

This is called the n-dimensional real vector space and is denoted $$\mathbb{R}^n$$. The metric induces a topology on $$\mathbb{R}^n$$, which is called the Euclidean topology.

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