To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product, i. e.

Geometrically, one studies the Euclidean plane ( 2 dimensions ) and Euclidean space ( 3 dimensions ).

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

It is simpler to see the notational equivalences between ordinary notation and bra-ket notation, so for now ; consider a vector A as an element of 3-d Euclidean space using the field of real numbers, symbolically stated as.

For example, in three-dimensional complex Euclidean space,

Structures analogous to those found in continuous geometries ( Euclidean plane, real projective space, etc.

Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body being modeled.

Different configurations or states of the body correspond to different regions in Euclidean space.

The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

Euclidean space itself is not compact since it is not bounded.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions.

A subset of Euclidean space in particular is called compact if it is closed and bounded.

That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.

; Euclidean space

For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.

Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.

The algebra M < sub > n </ sub >( C ) of n-by-n matrices over C becomes a C *- algebra if we consider matrices as operators on the Euclidean space, C < sup > n </ sup >, and use the operator norm ||.|| on matrices.

The convolution can be defined for functions on groups other than Euclidean space.

Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space.

The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.

This length function satisfies the required properties of a norm and is called the Euclidean norm on R < sup > n </ sup >.

( Many authors refer to R < sup > n </ sup > itself as Euclidean space, with the Euclidean structure being understood ).

The Euclidean topology turns out to be equivalent to the product topology on R < sup > n </ sup > considered as a product of n copies of the real line R ( with its standard topology ).

The simplest Riemannian manifold, consisting of R < sup > n </ sup > with a constant inner product, is essentially identical to Euclidean n-space itself.

So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID.

However, if there is no " obvious " Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

A Euclidean function on R is a function

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

Let R be a domain and f a Euclidean function on R. Then:

In Euclidean space R < sup > n </ sup >, or any convex subset of R < sup > n </ sup >, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

For nearby astronomical objects ( such as stars in our galaxy ) luminosity distance D < sub > L </ sub > is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean.

A dynamical system is the tuple, with a manifold ( locally a Banach space or Euclidean space ), the domain for time ( non-negative reals, the integers, ...) and f an evolution rule t → f < sup > t </ sup > ( with ) such that f < sup > t </ sup > is a diffeomorphism of the manifold to itself.

Real coordinate space together with this Euclidean structure is called Euclidean space and often denoted E < sup > n </ sup >.

The Euclidean structure makes E < sup > n </ sup > an inner product space ( in fact a Hilbert space ), a normed vector space, and a metric space.

The metric topology on E < sup > n </ sup > is called the Euclidean topology.

Although the problem is not known to be in NC, parallel algorithms with time superior to the Euclidean algorithm exist ; the best known deterministic algorithm is by Chor and Goldreich, which ( in the CRCW-PRAM model ) can solve the problem in O ( n / log n ) time with n < sup > 1 + ε </ sup > processors.

* Harmonic analysis on Euclidean spaces deals with properties of the Fourier transform on R < sup > n </ sup > that have no analog on general groups.

: More generally, any Euclidean space < sup > n </ sup > with the dot product is an inner product space

While the Möbius strip can be embedded in three-dimensional Euclidean space R < sup > 3 </ sup >, the Klein bottle cannot.

* The three-dimensional Euclidean space R < sup > 3 </ sup > with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra.

Thus Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.