In mathematics, a topological vector space is one of the basic structures investigated in functional analysis.
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In particular, they give conditions when functions with closed graphs are necessarily continuous. In mathematics, there are several results known as the "closed graph theorem".
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 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 by finitely many subsets of every fixed "size"
In functional analysis and related areas of mathematics, a barrelled space is a topological vector spaces (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 and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
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 maps from a bornological space into any locally convex spaces is continuous if and only if it a bounded linear operator.
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 theorem to hold for a wider class of linear maps. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
In functional analysis and related areas of mathematics, a topological vector spaces (TVS) is complete if every net that is Cauchy with respect to the space's canonical uniformity always converges to some point. Said differently, a TVS is complete if its canonical uniformity is complete. The canonical uniformity on a TVS (X, τ) is the unique translation-invariant uniformity that induces the topology τ on X. This notion of "TVS-completeness" depends only on vector subtraction and the topology of the TVS and so can be applied to all TVSs, including those whose topologies can not be defined in terms metrics. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms of completeness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness; a notion that is independent of any particular norm or metric.
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 D is bounded: in this case, the auxiliary normed space is span D with norm pD :=infx∈rD, r>0r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p-1
D(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.
In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X 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, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled if every bounded absorbing barrel is a neighborhood of the origin.
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, 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 X that has an associated vector bornology ℬ is called bornivorous and a bornivore if it absorbs every element of ℬ. If X is a topological vector space (TVS) then a subset S of X is bornivorous if it is bornivorous with respect to the von-Neumann bornology of X.
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.
In functional analysis and related areas of mathematics, an almost open linear map between topological vector spacess (TVSs) is a linear operator that satisfies a condition similar to, but weaker than, the condition of being an open map.
F. Riesz's theorem 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 locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.
