Espace tonneléEn analyse fonctionnelle et dans les domaines proches des mathématiques, les espaces tonnelés sont des espaces vectoriels topologiques où tout ensemble tonnelé - ou tonneau - de l'espace est un voisinage du vecteur nul. La raison principale de leur importance est qu'ils sont exactement ceux pour lesquels le théorème de Banach-Steinhaus s'applique. Nicolas Bourbaki a inventé des termes tels que « tonneau » ou espace « tonnelé » (à partir des tonneaux de vin) ainsi que les espaces « bornologiques ».
Partie bornée d'un espace vectoriel topologiqueEn analyse fonctionnelle et dans des domaines mathématiques reliés, une partie d'un espace vectoriel topologique est dite bornée (au sens de von Neumann) si tout voisinage du vecteur nul peut être dilaté de manière à contenir cette partie. Ce concept a été introduit par John von Neumann et Andreï Kolmogorov en 1935. Les parties bornées sont un moyen naturel de définir les (localement convexes) sur les deux espaces vectoriels d'une paire duale.
Bornological spaceIn 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.
Complete topological vector spaceIn 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. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces.
Auxiliary normed spaceIn 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 (as topological vector spaces and as normed spaces).
Quasi-complete spaceIn 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. Every quasi-complete TVS is sequentially complete. In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact. In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.
Infrabarrelled spaceIn functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood of the origin. If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarrelled. Every quasi-complete infrabarrelled space is barrelled. Every barrelled space is infrabarrelled.
Strong dual spaceIn 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.
Quasibarrelled spaceIn 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. A subset of a topological vector space (TVS) is called bornivorous if it absorbs all bounded subsets of ; that is, if for each bounded subset of there exists some scalar such that A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed.
Distinguished spaceIn 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 dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual. Suppose that is a locally convex space and let and denote the strong dual of (that is, the continuous dual space of endowed with the strong dual topology).
