In the mathematical 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. [1]
DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in ( Grothendieck 1954 ). Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If is a metrizable locally convex space and is a sequence of convex 0-neighborhoods in such that absorbs every strongly bounded set, then is a 0-neighborhood in (where is the continuous dual space of endowed with the strong dual topology). [2]
A locally convex topological vector space (TVS) is a DF-space, also written (DF)-space, if [1]
The strong dual space of a Fréchet space is a DF-space. [7]
However,
There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space. [10] There exist DF-spaces having closed vector subspaces that are not DF-spaces. [11]
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that
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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.
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 the 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, 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 are however quite different from Hilbert spaces, another generalization of finite-dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.
In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces and , the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map is continuous. When equipped with this topology, is denoted and called the projective tensor product of and .
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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.
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