Vector bornology

In mathematics, especially functional analysis, a bornology formula_1 on a vector space formula_2 over a field formula_3 where formula_4 has a bornology ℬformula_5, is called a vector bornology if formula_1 makes the vector space operations into bounded maps.
Definitions.
Prerequisits.
A on a set formula_2 is a collection formula_1 of subsets of formula_2 that satisfy all the following conditions:
Elements of the collection formula_1 are called or simply if formula_1 is understood. 
The pair formula_12 is called a or a .
A or of a bornology formula_1 is a subset formula_14 of formula_1 such that each element of formula_1 is a subset of some element of formula_17 Given a collection formula_18 of subsets of formula_19 the smallest bornology containing formula_18 is called the bornology generated by formula_21
If formula_12 and formula_23 are bornological sets then their on formula_24 is the bornology having as a base the collection of all sets of the form formula_25 where formula_26 and formula_27 
A subset of formula_24 is bounded in the product bornology if and only if its image under the canonical projections onto formula_2 and formula_30 are both bounded. 
If formula_12 and formula_23 are bornological sets then a function formula_33 is said to be a or a (with respect to these bornologies) if it maps formula_1-bounded subsets of formula_2 to formula_36-bounded subsets of formula_37 that is, if formula_38 
If in addition formula_39 is a bijection and formula_40 is also bounded then formula_39 is called a . 
Vector bornology.
Let formula_2 be a vector space over a field formula_4 where formula_4 has a bornology formula_45 
A bornology formula_1 on formula_2 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 formula_2 is a topological vector space (TVS) and formula_1 is a bornology on formula_19 then the following are equivalent: 
A vector bornology formula_1 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 formula_52 
And a vector bornology formula_1 is called if the only bounded vector subspace of formula_2 is the 0-dimensional trivial space formula_55 
Usually, formula_4 is either the real or complex numbers, in which case a vector bornology formula_1 on formula_2 will be called a if formula_1 has a base consisting of convex sets.
Characterizations.
Suppose that formula_2 is a topological vector space (TVS) over the field formula_5 of real or complex numbers and formula_1 is a bornology on formula_63
Then the following are equivalent:
Bornology on a topological vector space.
If formula_2 is a topological vector space (TVS) then the set of all bounded subsets of formula_2 from a vector bornology on formula_2 called the , the , or simply the of formula_2 and is referred to as .
In any locally convex TVS formula_19 the set of all closed bounded disks form a base for the usual bornology of formula_63
Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.
Topology induced by a vector bornology.
Suppose that formula_2 is a vector space over the field formula_4 of real or complex numbers and formula_1 is a vector bornology on formula_63
Let formula_74 denote all those subsets formula_75 of formula_2 that are convex, balanced, and bornivorous.
Then formula_74 forms a neighborhood basis at the origin for a locally convex TVS topology.
Examples.
Locally convex space of bounded functions.
Let formula_4 be the real or complex numbers (endowed with their usual bornologies), let formula_79 be a bounded structure, and let formula_80 denote the vector space of all locally bounded formula_4-valued maps on formula_82
For every formula_83 let formula_84 for all formula_85 where this defines a seminorm on formula_63
The locally convex TVS topology on formula_80 defined by the family of seminorms formula_88 is called the .
This topology makes formula_80 into a complete space.
Bornology of equicontinuity.
Let formula_90 be a topological space, formula_4 be the real or complex numbers, and let formula_92 denote the vector space of all continuous formula_4-valued maps on formula_82
The set of all equicontinuous subsets of formula_92 forms a vector bornology on formula_96