# Category: Functional analysis

Dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Schur's property
In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm.
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be
Korovkin approximation
In mathematics the Korovkin approximation is a convergence statement in which the approximation of a function is given by a certain sequence of functions. In practice a continuous function can be appr
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generaliz
Zeta function (operator)
The zeta function of a mathematical operator is a function defined as for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "
Lyapunov vector
In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been
Numerical methods in fluid mechanics
Fluid motion is governed by the Navier–Stokes equations, a set of coupled and nonlinearpartial differential equations derived from the basic laws of conservation of mass, momentumand energy. The unkno
Dirichlet space
In mathematics, the Dirichlet space on the domain (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space , for
Measure of non-compactness
In functional analysis, two measures of non-compactness are commonly used; these associate numbers to sets in such a way that compact sets all get the measure 0, and other sets get measures that are b
Wetzel's problem
In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a math
Energetic space
In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for th
Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set of -dimensional normed spaces. With this distance, the set of isometry classes of -
Infinite-dimensional optimization
In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an i
Weighted space
In functional analysis, a weighted space is a space of functions under a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the w
Riesz sequence
In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants such that for all sequences of scalars (an) in the ℓp space ℓ2. A Riesz sequence is ca
Density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of an
In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form: Th
Inductive tensor product
The finest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) separately continuous is called th
Operator (mathematics)
In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly the same space, sometimes required to be the same space).
Uniformly bounded representation
In mathematics, a uniformly bounded representation of a locally compact group on a Hilbert space is a homomorphism into the bounded invertible operators which is continuous for the strong operator top
Constructive quantum field theory
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requ
Band (order theory)
In mathematics, specifically in order theory and functional analysis, a band in a vector lattice is a subspace of that is solid and such that for all such that exists in we have The smallest band cont
Müntz–Szász theorem
The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadam
Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly sup
Strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topol
Bounded mean oscillation
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions
Bornology
In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bo
Multiplication operator
In operator theory, a multiplication operator is an operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, for
Kuratowski embedding
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement obviously holds for the empty space
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator
Normal cone (functional analysis)
In mathematics, specifically in order theory and functional analysis, if is a cone at the origin in a topological vector space such that and if is the neighborhood filter at the origin, then is called
Dirichlet algebra
In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on
Associative property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, asso
Schwartz topological vector space
In functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally boun
Degenerate bilinear form
In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v
Dual space
In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on , together with the vector space structure of pointwise addition
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first
Locally compact quantum group
In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors an
Banach limit
In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers :
Ordered algebra
In mathematics, an ordered algebra is an algebra over the real numbers with unit e together with an associated order such that e is positive (i.e. e ≥ 0), the product of any two positive elements is a
Sobolev spaces for planar domains
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems
Jordan operator algebra
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Ban
Polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a sub
Countably barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equic
Spectrum continuation analysis
Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier
Free probability
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is
Neumann series
A Neumann series is a mathematical series of the form where is an operator and its times repeated application. This generalizes the geometric series. The series is named after the mathematician Carl N
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in
Bs space
In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers or complex numbers such that is finite. The set of such sequences forms a normed
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinit
Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the mat
Distortion problem
In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Ba
Gelfand representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous fu
K-space (functional analysis)
In mathematics, more specifically in functional analysis, a K-space is an F-space such that every extension of F-spaces (or twisted sum) of the form is equivalent to the trivial onewhere is the real l
Archimedean ordered vector space
In mathematics, specifically in order theory, a binary relation on a vector space over the real or complex numbers is called Archimedean if for all whenever there exists some such that for all positiv
Gelfand–Shilov space
In the mathematical field of functional analysis, a Gelfand–Shilov space is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov .
Webbed space
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theore
F. Riesz's theorem
F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is loc
Montel space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Mo
Continuous embedding
In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almos
Regularly ordered
In mathematics, specifically in order theory and functional analysis, an ordered vector space is said to be regularly ordered and its order is called regular if is Archimedean ordered and the order du
Tensor product of Hilbert spaces
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilb
Uniform algebra
In functional analysis, a uniform algebra A on a compact Hausdorff topological space X is a closed (with respect to the uniform norm) subalgebra of the C*-algebra C(X) (the continuous complex-valued f
Scarborough criterion
The Scarborough criterion is used for satisfying convergence of a solution while solving linear equations using an iterative method.
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean sp
Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, m
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually
Functional determinant
In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a fi
Banach–Mazur theorem
In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named a
Star domain
In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the line segment from to lies in Thi
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theor
Topological homomorphism
In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept
Quasicontraction semigroup
In mathematical analysis, a C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0. Γ(t) is called a contraction semigroup
Riesz's lemma
Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may als
Orthogonal basis
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized,
Complemented subspace
In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (top
Invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some no
Gelfand–Naimark–Segal construction
In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain lin
Analytic semigroup
In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuo
Wiener series
In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Vol
Choquet theory
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. R
Volterra series
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for app
Coercive function
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the contextdifferent exact definitions of this idea are in use.
Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector sp
Pettis integral
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a me
Baire space
In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.According to the Baire category theorem, compact Hausdor
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If is a subs
Abstract L-space
In mathematics, specifically in order theory and functional analysis, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice whose norm is additive on the positive cone of
Local boundedness
In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that p
Ordered topological vector space
In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤
Extensions of symmetric operators
In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoi
Sherman–Takeda theorem
In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A. The t
Square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the in
Negacyclic convolution
In mathematics, negacyclic convolution is a convolution between two vectors a and b. It is also called skew circular convolution or wrapped convolution. It results from multiplication of a skew circul
Riesz space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word
James' space
In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerni
Algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior.
Solid set
In mathematics, specifically in order theory and functional analysis, a subset of a vector lattice is said to be solid and is called an ideal if for all and if then An ordered vector space whose order
Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers.
Composition operator
In mathematics, the composition operator with symbol is a linear operator defined by the rule where denotes function composition. The study of composition operators is covered by AMS category 47B33.
Refinement (category theory)
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differentia
Asplund space
In mathematics — specifically, in functional analysis — an Asplund space or strong differentiability space is a type of well-behaved Banach space. Asplund spaces were introduced in 1968 by the mathema
Bergman space
In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are suff
Extreme point
In mathematics, an extreme point of a convex set in a real or complex vector space is a point in which does not lie in any open line segment joining two points of In linear programming problems, an ex
List of functional analysis topics
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in pro
Markushevich basis
In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both complete and total.
Perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A cri
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it give
Saturated family
In mathematics, specifically in functional analysis, a family of subsets a topological vector space (TVS) is said to be saturated if contains a non-empty subset of and if for every the following condi
Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions defined on a set the non-negative number This norm is also called the supremum norm, the C
Choquet integral
A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but foun
Explicit algebraic stress model
The algebraic stress model arises in computational fluid dynamics. Two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent
Sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an e
Spectral theory of compact operators
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closur
Orthogonal complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V th
Lyapunov–Schmidt reduction
In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permi
The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like
Antiunitary operator
In mathematics, an antiunitary transformation, is a bijective antilinear map between two complex Hilbert spaces such that for all and in , where the horizontal bar represents the complex conjugate. If
Locally convex vector lattice
In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in
Mollifier
In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions
Complementarity theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topolog
Geometric quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which ther
Operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linea
Convolution power
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if is a function on Euclidean space Rd and is a natural number, then the convolution power is defined
Harmonic spectrum
A harmonic spectrum is a spectrum containing only frequency components whose frequencies are whole number multiples of the fundamental frequency; such frequencies are known as harmonics. "The individu
Compact embedding
In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology a
Projective tensor product
The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projec
Conjugate index
In mathematics, two real numbers are called conjugate indices (or Hölder conjugates) if Formally, we also define as conjugate to and vice versa. Conjugate indices are used in Hölder's inequality. If a
Bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a n
Kernel (linear algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L
Circular convolution
Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution ar
Quasi-complete space
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance f
Spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usual
Operator ideal
In functional analysis, a branch of mathematics, an operator ideal is a special kind of class of continuous linear operators between Banach spaces. If an operator belongs to an operator ideal , then f
High-dimensional statistics
In statistical theory, the field of high-dimensional statistics studies data whose dimension is larger than typically considered in classical multivariate analysis. The area arose owing to the emergen
Unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be use
Stechkin's lemma
In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq norm of the tail of a sequence, when the whole sequence is known to have f
Auxiliary normed space
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used i
International Workshop on Operator Theory and its Applications
International Workshop on Operator Theory and its Applications (IWOTA) was started in 1981 to bring together mathematicians and engineers working in operator theoretic side of functional analysis and
Lower envelope
In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value at every point is the minimum of the values of
Strongly positive bilinear form
A bilinear form, a(•,•) whose arguments are elements of normed vector space V is a strongly positive bilinear form if and only if there exists a constant, c>0, such that for all where is the norm on V
Topological vector lattice
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice
Order bound dual
In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space is the set of all linear functionals on that map order intervals, which are sets o
Hypercyclic operator
In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0,
Countably quasi-barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is aga
Cyclic vector
An operator A on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector f if the vectors f, Af, A2f,... span H. Equivalently, f is a cyclic vector for A in case the set of all v
Disk algebra
In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ : D → , (where D is the open unit disk in the c
Hamburger moment problem
In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m0, m1, m2, ...), does there exist a positive Borel measure μ (for instance
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod
In mathematics, a subset of a linear space is radial at a given point if for every there exists a real such that for every Geometrically, this means is radial at if for every there is some (non-degene
Schröder–Bernstein theorems for operator algebras
The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is t
Operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of map
Supporting hyperplane
In geometry, a supporting hyperplane of a set in Euclidean space is a hyperplane that has both of the following two properties: * is entirely contained in one of the two closed half-spaces bounded by
Beppo-Levi space
In functional analysis, a branch of mathematics, a Beppo Levi space, named after Beppo Levi, is a certain space of generalized functions. In the following, D′ is the space of distributions, S′ is the
Compact convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
Prevalent and shy sets
In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the transla
Brauner space
In functional analysis and related areas of mathematics a Brauner space is a complete compactly generated locally convex space having a sequence of compact sets such that every other compact set is co
Discontinuous linear map
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see li
Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is
Abstract m-space
In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice whose norm satisfies for all x and y in the positive cone of X. We say that
Fréchet lattice
In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of to
C0-semigroup
In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar
Exposed point
In mathematics, an exposed point of a convex set is a point at which some continuous linear functional attains its strict maximum over . Such a functional is then said to expose . There can be many ex
Totally bounded space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered b
Barrier cone
In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar s
Mackey–Arens theorem
The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous d
Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with
Ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Kantorovich theorem
The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar t
Uniformly Cauchy sequence
In mathematics, a sequence of functions from a set S to a metric space M is said to be uniformly Cauchy if: * For all , there exists such that for all : whenever . Another way of saying this is that
Order summable
In mathematics, specifically in order theory and functional analysis, a sequence of positive elements in a preordered vector space (that is, for all ) is called order summable if exists in . For any ,
Cocompact embedding
In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function that expresses how the shape of one is modified b
Abstract differential equation
In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach spa
Nuclear C*-algebra
In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra such that the injective and projective C*-cross norms on are the same for every C*-algebra This property was firs
Helly space
In mathematics, and particularly functional analysis, the Helly space, named after Eduard Helly, consists of all monotonically increasing functions ƒ : [0,1] → [0,1], where [0,1] denotes the closed in
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize norme
Total subset
In mathematics, more specifically in functional analysis, a subset of a topological vector space is said to be a total subset of if the linear span of is a dense subset of This condition arises freque
Bochner measurable function
In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable
Triebel–Lizorkin space
In the mathematical discipline known as functional analysis, a Triebel–Lizorkin space is a generalization of many standard function spaces such as Lp spaces and Sobolev spaces. It is named after (born
Injective tensor product
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product
Bred vector
In applied mathematics, bred vectors are perturbations, related to Lyapunov vectors, that capture fast-growing dynamical instabilities of the solution of a numerical model. They are used, for example,
Order convergence
In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice is order convergent if it contains an order bounded subset (that is, is contained in
System of imprimitivity
The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the
C space
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences of real numbers or complex numbers. When equipped with the uniform norm: the sp
Workshop on Numerical Ranges and Numerical Radii
Workshop on Numerical Ranges and Numerical Radii (WONRA) is a biennial workshop series on numerical ranges and numerical radii which began in 1992.
Lattice disjoint
In mathematics, specifically in order theory and functional analysis, two elements x and y of a vector lattice X are lattice disjoint or simply disjoint if , in which case we write , where the absolut
Fusion frame
In mathematics, a fusion frame of a vector space is a natural extension of a frame. It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from
Complete topological vector space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each
Vector measure
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes
Normed vector lattice
In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are i
Positive linear functional
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements that is it holds that In o
Direct integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of v
Densely defined operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator th
State (functional analysis)
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which rep
Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose
Variance function
In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obey
CCR and CAR algebras
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermi
Continuous linear extension
In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space by first defining a linear transformation on a dense subset of and then extending to
Order complete
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (mea
L-semi-inner product
In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal wit
Supporting functional
In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.
Dual system
In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is
Topologies on spaces of linear maps
In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can gi
Quasitrace
In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if e
Orthogonal functions
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be th
Borel graph theorem
In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz. The Borel graph theorem shows that the closed graph theorem is valid for l
Hölder condition
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α > 0, such tha
Onsager–Machlup function
The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to th
Bounded deformation
In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric
Spectral set
In operator theory, a set is said to be a spectral set for a (possibly unbounded) linear operator on a Banach space if the spectrum of is in and von-Neumann's inequality holds for on - i.e. for all ra
Hyers–Ulam–Rassias stability
The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms. In the next year, Donald H. Hyers gave a par
Smith space
In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space having a universal compact set, i.e. a compact set whic
Order dual (functional analysis)
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear
Weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The resu
Weakly measurable function
In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable func
List of mathematic operators
In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differe
Free independence
In the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu. The definition of free independence is parallel to the classical definition of indepen
Modes of variation
In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variatio
Bochner space
In mathematics, Bochner spaces are a generalization of the concept of spaces to functions whose values lie in a Banach space which is not necessarily the space or of real or complex numbers. The space
Colombeau algebra
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is
Partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued funct
Weak order unit
In mathematics, specifically in order theory and functional analysis, an element of a vector lattice is called a weak order unit in if and also for all
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If
Stone–von Neumann theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between positi
Banach lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice (X,‖·‖) is a complete normed vector space with a lattice order, such that for all x, y ∈ X, the implication
Spherically complete field
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is
Liouville space
In the mathematical physics of quantum mechanics, Liouville space, also known as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Sch
Glossary of functional analysis
This is a glossary for the terminology in a mathematical field of functional analysis. Throughout the article, unless stated otherwise, the base field of a vector space is the field of real numbers or
Orlicz sequence space
In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz
Orthonormal function system
An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. See basis (linear algebra), Fourier analysis, square-integrable, Hilbert space for more.
Envelope (category theory)
In Category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech c
Sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of
Unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the tran
Cauchy–Rassias stability
A classical problem of Stanislaw Ulam in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation E must be close to an
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs
Cylindrical σ-algebra
In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or p
Symmetric convolution
In mathematics, symmetric convolution is a special subset of convolution operations in which the convolution kernel is symmetric across its zero point. Many common convolution-based processes such as
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to
Cone-saturated
In mathematics, specifically in order theory and functional analysis, if is a cone at 0 in a vector space such that then a subset is said to be -saturated if where Given a subset the -saturated hull o
Order topology (functional analysis)
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space is the finest locally convex topological vector space (TVS) topology on for which ev
Closed graph property
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : X → Y between topological spaces has a closed graph if its graph is a closed s
Beta-dual space
In functional analysis and related areas of mathematics, the beta-dual or β-dual is a certain linear subspace of the algebraic dual of a sequence space.
Transpose of a linear map
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic
Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor call
Free convolution
Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately
Infrabarrelled space
In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded absorbing barrel is
Linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers
Sublinear function
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function w
Spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topologica
Predual
In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of th
Christoffel–Darboux formula
In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel and Jean Gaston Darboux. It states that where fj(x) is th
Compression (functional analysis)
In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator , where is the orthogonal projection onto K. This is a natural way to obtain an operat
Functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square roo
Operator topologies
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.
Positive linear operator
In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space into a preordered vector space is a linear operator on into such that for all posit
DF-space
In the field of functional analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector sp
Trigonometric moment problem
In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval [0, 2π] such that In other
Unit ball
No description available.
Quasi-interior point
In mathematics, specifically in order theory and functional analysis, an element of an ordered topological vector space is called a quasi-interior point of the positive cone of if and if the order int