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Formally and Hilbert
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 ).
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 bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,

Formally and space
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, 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, an ultrametric space is a set of points with an associated distance function ( also called a metric )
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 consists
Formally it consists of a short lyric composed in a single metre with a single rhyme throughout.
Formally, a generalised self reproducing system consists
Formally it consists of three groups of four measure phrases of three to four chords repeated four times each, ABC :|| ABC, which then merges with the next movement, " Floe " with the entrance of the horns.
Formally, a deterministic Büchi automaton is a tuple A = ( Q, Σ, δ, q < sub > 0 </ sub >, F ) that consists of the following components:
Formally, given two categories C and D, an equivalence of categories consists of a functor F: C → D, a functor G: D → C, and two natural isomorphisms ε: FG → I < sub > D </ sub > and η: I < sub > C </ sub >→ GF.
Formally, a semigroupoid consists of:
Formally, a bicategory B consists of:
Formally, Mbenga – Mbuti Pygmy music consists of at most only four parts, and can be described as an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.
Formally, a monad consists of a type constructor M and two operations, bind and return.
Formally, the game ( coalitional game ) consists of a finite set of players, called the grand coalition, and a characteristic function denotes the power set of .</ ref > from the set of all possible coalitions of players to a set of payments that satisfies.
Formally, he says, the piece consists of two parts of nearly equal length, the end of the first section being bars 24 – 28 ( p. 77 ).
Formally, a parsing expression grammar consists of:
Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form, with being called pattern graph ( or left-hand side ) and being called replacement graph ( or right-hand side of the rule ).
Formally, an institution consists of
Formally, a signed graph Σ is a pair ( G, σ ) that consists of a graph G = ( V, E ) and a sign mapping or signature σ from E to the sign group
Formally, the right Kan extension of along consists of a functor and a natural transformation which is couniversal with respect to the specification, in the sense that for any functor and natural transformation, a unique natural transformation is defined and fits into a commutative diagram

Formally and H
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, the index of H in G is defined as the number of cosets of H in G. ( The number of left cosets of H in G is always equal to the number of right cosets.
Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: HH ( called the antipode ) such that the following diagram commutes:
Formally, given a G-bundle B and a map H → G ( which need not be an inclusion ),

Formally and together
Formally, it was ceded to Russia in the Treaty of Nystad in 1721, together with Swedish Estonia and Swedish Ingria.
Formally, a ring is an Abelian group ( R, +), together with a second binary operation * such that for all a, b and c in R,
Formally, an absolute coequalizer of a pair in a category C is a coequalizer as defined above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.
Formally known as the World Commission on Environment and Development ( WCED ), the Brundtland Commission's mission is to unite countries to pursue sustainable development together.

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