In mathematics, especially functional analysis, a bornology on a vector space over a field where has a bornology â¬<sub></sub>, is called a vector bornology if makes the vector space operations into bounded maps.
A on a set is a collection of subsets of that satisfy all the following conditions:
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 bornology generated by
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 .
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.).
If is a vector space and is a bornology on then the following are equivalent:
A vector bornology is called a if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then And a vector bornology is called if the only bounded vector subspace of is the 0-dimensional trivial space
Usually, is either the real or complex numbers, in which case a vector bornology on will be called a if has a base consisting of convex sets.
Suppose that is a vector space over the field of real or complex numbers and is a bornology on Then the following are equivalent:
If is a topological vector space then the set of all bounded subsets of from a vector bornology on called the , the , or simply the of and is referred to as . In any locally convex topological vector space the set of all closed bounded disks form a base for the usual bornology of
Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.
Suppose that is a vector space over the field of real or complex numbers and is a vector bornology on Let denote all those subsets of that are convex, balanced, and bornivorous. Then forms a neighborhood basis at the origin for a locally convex topological vector space topology.
Let be the real or complex numbers (endowed with their usual bornologies), let be a bounded structure, and let denote the vector space of all locally bounded -valued maps on For every let for all where this defines a seminorm on The locally convex topological vector space topology on defined by the family of seminorms is called the . This topology makes into a complete space.
Let be a topological space, be the real or complex numbers, and let denote the vector space of all continuous -valued maps on The set of all equicontinuous subsets of forms a vector bornology on