Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, these reside in a complex separable Hilbert space-variously called the " state space " or the " associated Hilbert space " of the system-that is well defined up to a complex number of norm 1 ( the phase factor ).

Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space.

Formally, the question of whether the universe is infinite or finite is whether it is an unbounded or bounded metric space.

Formally, we start with a metric space M and a subset X.

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > ( Some may also prefer the alternative signature < tt >(+,−,−,−)</ tt >; in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.

Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.

Formally, a ringed space ( X, O < sub > X </ sub >) is a topological space X together with a sheaf of rings O < sub > X </ sub > on X.

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion

Formally, a frame on a homogeneous space G / H consists of a point in the tautological bundle G → G / H.

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

Formally, a state space can be defined as a tuple where:

Formally, rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.

Formally, a complex projective space is the space of complex lines through the origin of an ( n + 1 )- dimensional complex vector space.

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, the system is said to have memory.

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally, Φ = kx − ωt is the phase.

Formally speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).

Formally, one has an implication, not an equivalence, so the solution set may get larger.

Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.

Formally, the weighted mean of a non-empty set of data

Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

Formally, Gaussian integers are the set

Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.

Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.

Formally, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.

Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.

Formally, starting with a linearly independent set of vectors

Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).

Formally, a constraint satisfaction problem is defined as a triple, where is a set of variables, is a domain of values, and is a set of constraints.