Bornological space

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 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. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after borné, the French word for "bounded". Bornology A on a set is a collection of subsets of that satisfy all the following conditions: covers that is, ; is stable under inclusions; that is, if and then ; is stable under finite unions; that is, if then ; Elements of the collection are called or simply if is understood. The pair is called a or a . A or of a bornology is a subset of such that each element of is a subset of some element of Given a collection of subsets of the smallest bornology containing is called the If and are bornological sets then their on is the bornology having as a base the collection of all sets of the form where and A subset of is bounded in the product bornology if and only if its image under the canonical projections onto and are both bounded. If and are bornological sets then a function is said to be a or a (with respect to these bornologies) if it maps -bounded subsets of to -bounded subsets of that is, if If in addition is a bijection and is also bounded then is called a . Vector bornology Let be a vector space over a field where has a bornology A bornology on is called a if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).