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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 spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for, in fact a Hilbert spaceIn mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the F, in a natural way. The inner product, also called the dot product, of x = (x₁,...,x_n) and y = (y₁,...,y_n) is given by

See also: Euclidean geometryEuclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry someti.