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 that 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 Mackey. The name was coined by Bourbaki after , the French word for "bounded".
Bornologies and bounded maps.
A on a set formula_1 is a collection formula_2 of subsets of formula_1 that satisfy all the following conditions:
Elements of the collection formula_2 are called or simply if formula_2 is understood. 
The pair formula_6 is called a or a .
A or of a bornology formula_2 is a subset formula_8 of formula_2 such that each element of formula_2 is a subset of some element of formula_11 Given a collection formula_12 of subsets of formula_13 the smallest bornology containing formula_12 is called the bornology generated by formula_15
If formula_6 and formula_17 are bornological sets then their on formula_18 is the bornology having as a base the collection of all sets of the form formula_19 where formula_20 and formula_21 
A subset of formula_18 is bounded in the product bornology if and only if its image under the canonical projections onto formula_1 and formula_24 are both bounded. 
Bounded maps.
If formula_6 and formula_17 are bornological sets then a function formula_27 is said to be a or a (with respect to these bornologies) if it maps formula_2-bounded subsets of formula_1 to formula_30-bounded subsets of formula_31 that is, if formula_32 
If in addition formula_33 is a bijection and formula_34 is also bounded then formula_33 is called a .
Vector bornologies.
Let formula_1 be a vector space over a field formula_37 where formula_37 has a bornology formula_39 
A bornology formula_2 on formula_1 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_1 is a topological vector space (TVS) and formula_2 is a bornology on formula_13 then the following are equivalent: 
A vector bornology formula_2 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_46 
And a vector bornology formula_2 is called if the only bounded vector subspace of formula_1 is the 0-dimensional trivial space formula_49 
Usually, formula_37 is either the real or complex numbers, in which case a vector bornology formula_2 on formula_1 will be called a if formula_2 has a base consisting of convex sets. 
Bornivorous subsets.
A subset formula_54 of formula_1 is called and a if it absorbs every bounded set.
In a vector bornology, formula_54 is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology formula_54 is bornivorous if it absorbs every bounded disk.
Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.
Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.
Mackey convergence.
A sequence formula_58 in a TVS formula_1 is said to be if there exists a sequence of positive real numbers formula_60 diverging to formula_61 such that formula_62 converges to formula_63 in formula_64
Bornology of a topological vector space.
Every topological vector space formula_13 at least on a non discrete valued field gives a bornology on formula_1 by defining a subset formula_67 to be bounded (or von-Neumann bounded), if and only if for all open sets formula_68 containing zero there exists a formula_69 with formula_70 
If formula_1 is a locally convex topological vector space then formula_67 is bounded if and only if all continuous semi-norms on formula_1 are bounded on formula_74
The set of all bounded subsets of a topological vector space formula_1 is called or 
If formula_1 is a locally convex topological vector space, then an absorbing disk formula_77 in formula_1 is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).
Induced topology.
If formula_2 is a convex vector bornology on a vector space formula_13 then the collection formula_81 of all convex balanced subsets of formula_1 that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on formula_1 called .
If formula_84 is a TVS then is the vector space formula_1 endowed with the locally convex topology induced by the von Neumann bornology of formula_86
Bornological spaces.
In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.
Quasi-bornological spaces.
Quasi-bornological spaces where introduced by S. Iyahen in 1968.
A topological vector space (TVS) formula_84 with a continuous dual formula_88 is called a if any of the following equivalent conditions holds:
Every pseudometrizable TVS is quasi-bornological. 
A TVS formula_84 in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space. 
If formula_1 is a quasi-bornological TVS then the finest locally convex topology on formula_1 that is coarser than formula_92 makes formula_1 into a locally convex bornological space.
Bornological space.
Note that every locally convex quasi-bornological space is bornological but there exist bornological spaces that are quasi-bornological.
A topological vector space (TVS) formula_84 with a continuous dual formula_88 is called a if it is locally convex and any of the following equivalent conditions holds:
If formula_1 is a Hausdorff locally convex space then we may add to this list:
Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous, where recall that a linear operator is sequentially continuous if and only if it is sequentially continuous at the origin. 
Thus for linear maps from a bornological space into a locally convex space, continuity is equivalent to sequential continuity at the origin. More generally, we even have the following:
Sufficient conditions.
As a consequent of the Mackey-Ulam theorem, "for all practical purposes, the product of bornological spaces 
is bornological."
The following topological vector spaces are all bornological:
There exists a bornological LB-space whose strong bidual is bornological. 
A closed vector subspace of a locally convex bornological space is not necessarily bornological. 
There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological. 
Bornological spaces need not be barrelled and barrelled spaces need not be bornological. Because every locally convex ultrabornological space is barrelled, it follows that a bornological space is not necessarily ultrabornological.
Ultrabornological spaces.
A disk in a topological vector space formula_1 is called if it absorbs all Banach disks.
If formula_1 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 if any of the following equivalent conditions hold:
Properties.
The finite product of ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological.