# Category: Topological vector spaces

Bounded set (topological vector space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to inclu
Nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces
Category of topological vector spaces
In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category be
Bochner integral
In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functi
Almost open map
In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open m
Differentiable vector–valued functions from Euclidean space
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domain
Biorthogonal system
In mathematics, a biorthogonal system is a pair of indexed families of vectors such thatwhere and form a pair of topological vector spaces that are in duality, is a bilinear mapping and is the Kroneck
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows
List of vector spaces in mathematics
This is a list of vector spaces in abstract mathematics, by Wikipedia page. * Banach space * Besov space * Bochner space * Dual space * Euclidean space * Fock space * Fréchet space * Hardy spa
Mackey topology
In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual
Metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM
Barrelled set
In functional analysis, a subset of a topological vector space (TVS) is called a barrel or a barrelled set if it is closed convex balanced and absorbing. Barrelled sets play an important role in the d
Bornivorous set
In functional analysis, a subset of a real or complex vector space that has an associated vector bornology is called bornivorous and a bornivore if it absorbs every element of If is a topological vect
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
FK-space
In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces
Quasibarrelled space
In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the orig
Real coordinate space
In mathematics, the real coordinate space of dimension n, denoted Rn (/ɑːrˈɛn/ ar-EN) or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. With componen
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are
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
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
Quasi-ultrabarrelled space
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.A topologi
Distinguished space
In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong
Ptak space
A locally convex topological vector space (TVS) is B-complete or a Ptak space if every subspace is closed in the weak-* topology on (i.e. or ) whenever is closed in (when is given the subspace topolog
BK-space
In functional analysis and related areas of mathematics, a BK-space or Banach coordinate space is a sequence space endowed with a suitable norm to turn it into a Banach space. All BK-spaces are normab
Schauder fixed-point theorem
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if is a nonempty convex closed sub
Integral linear operator
An integral bilinear form is a bilinear functional that belongs to the continuous dual space of , the injective tensor product of the locally convex topological vector spaces (TVSs) X and Y. An integr
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
Differentiation in Fréchet spaces
In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is
Barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourho
Crinkled arc
In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider where is a Hilbert s
Fundamental theorem of Hilbert spaces
In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space t
Vector bornology
In mathematics, especially functional analysis, a bornology on a vector space over a field where has a bornology ℬ, is called a vector bornology if makes the vector space operations into bounded maps.
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 ≤
Cylinder set measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian c
Vector-valued Hahn–Banach theorems
In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are alwa
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
Complex coordinate space
In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers. It is denoted , and is the n-fold Cartesian product of the comple
Functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that a
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector sp
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
LF-space
In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that
Mackey space
In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest to
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
FK-AK space
In functional analysis and related areas of mathematics an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis.
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
Topological tensor product
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved th
Ultrabornological space
In functional analysis, a topological vector space (TVS) is called ultrabornological if every bounded linear operator from into another TVS is necessarily continuous. A general version of the closed g
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
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.
Ultrabarrelled space
In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin.
Indefinite inner product space
In mathematics, in the field of functional analysis, an indefinite inner product space is an infinite-dimensional complex vector space equipped with both an indefinite inner product and a positive sem
Semi-Hilbert space
In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather t
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
Topological algebra
In mathematics, a topological algebra is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness
Infinite-dimensional vector function
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in mos
Schwartz space
In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on t
Nash–Moser theorem
In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on B
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
Krein–Smulian theorem
In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark
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
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
Krein–Milman theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem — A comp
Total set
In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals with the property that if a vector satisfies for all then is the zero vector. In a mor
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
LB-space
In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a dire
Gateaux derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician w
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space , such that the functional sending an operator to
Hypocontinuous bilinear map
In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that ar
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
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
Nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective
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.
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
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