Euclidean space

Euclidean space is the usual n-dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n, n-dimensional Euclidean space is the set Rⁿ (where R is the set of real numbers) together with the distance function obtained by defining the distance between two points (x₁, ..., x_n) and (y₁, ...,y_n) to be the square root of Σ (x_i-y_i)², where the sum is over i = 1, ..., n. This distance function is based on the Pythagorean Theorem and is called the Euclidean metric.

The term "n-dimensional Euclidean space" is usually abbreviated to "Euclidean n-space", or even just "n-space". Euclidean n-space is denoted by Eⁿ, although Rⁿ is also used (with the metric being understood). E² is called the Euclidean plane.

By definition, Eⁿ is a metric space, and is therefore also a topological space. It is the prototypical example of an n-manifold, and is in fact a differentiable n-manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Eⁿ is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces.

Much could be said about the topology of Eⁿ, but that will have to wait until a later revision of this article. One important result, Brouwer's invariance of domain, is that any subset of Eⁿ which is homeomorphic to an open subset of Eⁿ is itself open. An immediate consequence of this is that E^m is not homeomorphic to Eⁿ if m ≠ n -- an intuitively "obvious" result which is nonetheless not easy to prove.

Euclidean n-space can also be considered as an n-dimensional real vector space, in fact a Hilbert space, in a natural way. The inner product of x = (x₁,...,x_n) and y = (y₁,...,y_n) is given by

x · y = x₁y₁ + ... + x_ny_n.