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.
Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
For any set and scalar let
Given a topological vector space (TVS) over a field a subset of is called von Neumann bounded or just bounded in if any of the following equivalent conditions are satisfied:
while if is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include:
while if is a seminormed space with seminorm (note that every normed space is a seminormed space and every norm is a seminorm), then this list may be extended to include:
while if is a vector subspace of the TVS then this list may be extended to include:
A subset that is not bounded is called unbounded.
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of
A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some The set of all bounded subsets of trivially forms a fundamental system of bounded sets of
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.
Let be any topological vector space (TVS) (not necessarily Hausdorff or locally convex).
The definition of bounded sets can be generalized to topological modules. A subset of a topological module over a topological ring is bounded if for any neighborhood of there exists a neighborhood of such that
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
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. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In functional analysis, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all in
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 normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S 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.
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.
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 linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space is a set such that for all scalars satisfying
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced, in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that 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 subset is the polar of but lies in .
In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
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, 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 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 other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
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 is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.
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 if the disk is bounded: in this case, the auxiliary normed space is with norm The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.
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 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 topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, has the strong dual topology, or may be written.
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.
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.