Formally, let X be any scheme and S be a sheaf of graded-algebras ( the definition of which is similar to the definition of-modules on a locally ringed space ): that is, a sheaf with a direct sum decomposition

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 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, 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, an nth order approximation is one where the order of magnitude of the error is at most, or in terms of big O notation, the error is

Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

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, the ith row, jth column element of A < sup > T </ sup > is the jth row, ith column element of A:

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, 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 outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.

Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is

Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

Formally, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

Formally, a Lie superalgebra is a ( nonassociative ) Z < sub > 2 </ sub >- graded algebra, or superalgebra, over a commutative ring ( typically R or C ) whose product, called the Lie superbracket or supercommutator, satisfies the two conditions ( analogs of the usual Lie algebra axioms, with grading ):