Bornologies and Functional Analysis: Introductory course on the theory of duality topology-bornology and its use in functional analysis. Bornologies and functional analysis [electronic resource]: introductory course on the theory of duality topology-bornology and its use in functional analysis. : Bornologies and functional analysis, Volume Introductory course on the theory of duality topology-bornology and its use in functional.
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H Hogbe-Nlend (Author of Bornologies And Functional Analysis)
In mathematicsparticularly in functional analysisa 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 functionsin the same way that a topological space possesses the minimum amount of bornologgies needed to address questions of continuity. Bornological spaces were first studied by Mackey.
Elements of the collection B are usually called bounded sets.
A base of the bornology B is a subset B 0 of B such that each element of B is a subset of gunctional element of B 0. If B 1 and B 2 are two bornologies over the spaces X and Yanalysiw, and if f: If X is a vector space over a field K then a vector bornology on X is a bornology B on X that is stable under vector addition, scalar multiplication, and the formation of balanced hulls i.
If in addition B is stable under the formation of convex hulls i. And if the only bounded subspace of X is the trivial subspace i.
A subset A of X is called bornivorous if it absorbs every bounded set. In a vector bornology, A is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology A is bornollogies if it absorbs every bounded disk. The set of all bounded subsets of X is called the bornology or the Von-Neumann bornology of X.
Suppose that we start with a vector space X and convex vector bornology B on X.
Bornologies and functional analysis – CERN Document Server
If we let T denote the collection of all sets that are convex, balanced, and bornivorous then T forms neighborhood basis at 0 for a locally convex topology on X that is compatible with the vector space structure of X. In functional analysis, a bornological space is a locally convex topological vector space whose topology can be bornolofies from its bornology in a natural way.
Suppose that X is a topological vector space. Then we say that a subset D of X is a disk if it is convex and balanced.
A basis of neighborhoods of 0 of this space consists of all sets of the form r D where r ranges over all positive real numbers. A bounded disk in X for which X D is a Banach space is called a Banach diskinfracompleteor a bounded completant.
Suppose that X is a locally convex Hausdorff space. A disk in a topological vector space X is called infrabornivorous if it absorbs all Banach disks. If X is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks. A locally convex space is called ultrabornological if any of the following conditions hold:.
From Wikipedia, the free encyclopedia. Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point. Retrieved from ” https: Views Read Edit View history.