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Page "Compact space" ¶ 8
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Formally and topological
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, 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 and space
Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
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 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, an ultrametric space is a set of points with an associated distance function ( also called a metric )
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 and X
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.
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,
Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.
Formally styled as " Excelentísimo e Ilustrísimo Señor Profesor Doctor Don N, Rector Magnífico de la Universidad de X " ( Most Excellent and Illustrious Lord Professor Doctor Don N, Rector Magnificus of the University of X ), it is an office of high dignity within Spanish society, usually being highly respected.
Formally, a statistic s is a measurable function of X ; thus, a statistic s is evaluated on a random variable X, taking the value s ( X ), which is itself a random variable.
Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.
Formally, the mutual information of two discrete random variables X and Y can be defined as:
Formally, we begin by considering some family of distributions for a random variable X, that is indexed by some θ.
Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.
Formally, sending X to and acting as the identity on maps is a functor ( indeed, an involution ).
Formally, f < sub > X, Y </ sub >( x, y ) is the probability density function of ( X, Y ) with respect to the product measure on the respective supports of X and Y.

Formally and is
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.

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