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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.
A subset B0 of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1⊆Bi for all i = 0, 1, .... In this case, is called a defining sequence for B0. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin. [1]
A locally convex quasi-ultrabarrelled space is quasi-barrelled. [1]
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled. [1]
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that
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 space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
In functional analysis and related areas of mathematics, a barrelled space is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by Bourbaki (1950).
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.
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 include the set. A set that is not bounded is called unbounded.
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 Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.
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 of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.
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 X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
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 bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is becausepg 9 the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra.
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual is bijective. If this map is also an isomorphism of TVSs then it is called reflexive.
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 spaces. They play a considerable part in the theory of topological tensor products.
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 graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck.
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 for non-metrizable TVSs.
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 origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
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 bounded subsets. These spaces were introduced by Alexander Grothendieck.
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 are contained in the weak-* closure of some bounded subset of the bidual.
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 equicontinuous. This property is a generalization of barrelled spaces.
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 vector space (TVS) then a subset of is bornivorous if it is bornivorous with respect to the von-Neumann bornology of .
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 again equicontinuous. This property is a generalization of quasibarrelled spaces.
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.
