Ultrabornological space

In functional analysis, a topological vector space (TVS) is called ultrabornological if every bounded linear operator from 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 (Grothendieck [1955, p. 17] "espace du type (β)"). Let be a topological vector space (TVS). A disk is a convex and balanced set. A disk in a TVS is called bornivorous if it absorbs every bounded subset of A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks. A disk in a TVS is called infrabornivorous if it satisfies any of the following equivalent conditions: absorbs every Banach disks in while if locally convex then we may add to this list: the gauge of is an infrabounded map; while if locally convex and Hausdorff then we may add to this list: absorbs all compact disks; that is, is "compactivorious". A TVS is ultrabornological if it satisfies any of the following equivalent conditions: every infrabornivorous disk in is a neighborhood of the origin; while if is a locally convex space then we may add to this list: every bounded linear operator from into a complete metrizable TVS is necessarily continuous; every infrabornivorous disk is a neighborhood of 0; be the inductive limit of the spaces as D varies over all compact disks in ; a seminorm on that is bounded on each Banach disk is necessarily continuous; for every locally convex space and every linear map if is bounded on each Banach disk then is continuous; for every Banach space and every linear map if is bounded on each Banach disk then is continuous. while if is a Hausdorff locally convex space then we may add to this list: is an inductive limit of Banach spaces; Every locally convex ultrabornological space is barrelled, quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.