Euclidean spaceEuclid. 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, ..., xn) and (y1, ...,yn) to be the square root of Σ (xi-yi)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, also called the dot product, of x = (x1,...,xn) and y = (y1,...,yn) is given by