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 E ⁿ, although Rⁿ is also used (with the metric being understood). E ² is called the Euclidean plane.

By definition, E ⁿ 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 E ⁿ 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 E ⁿ, 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 E ⁿ which is homeomorphic to an open subset of E ⁿ is itself open. An immediate consequence of this is that E ^m is not homeomorphic to E ⁿ 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.

See also: Euclidean geometry.