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 functions, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces were first studied by Mackey.
The name was coined by Bourbaki after, the French word for "bounded".

Bornologies and bounded maps

A bornology on a set is a collection of subsets of such that
Elements of the collection are usually called -bounded or simply bounded sets.
The pair is called a bounded structure or a bornological set.
A base of the bornology' is a subset of such that each element of is a subset of an element of.

Bounded maps

If and are two bornologies over the spaces and, respectively, and if is a function, then we say that is a locally bounded map or a bounded map if it maps -bounded sets in to -bounded sets in.
If in addition is a bijection and is also bounded then we say that is a bornological isomorphism.

Vector bornologies

If is a vector space over a field then a vector bornology on is a bornology on that is stable under vector addition, scalar multiplication, and the formation of balanced hulls.
If is a topological vector space and is a bornology on, then the following are equivalent:
  1. is a vector bornology;
  2. finite sums and balanced hulls of -bounded sets are -bounded;
  3. the scalar multiplication map defined by and the addition map defined by, are both bounded when their domains carry their product bornologies.
If in addition is stable under the formation of convex hulls then is called a convex vector bornology.
And if the only bounded subspace of is the trivial subspace then it is called separated.
A subset of is called bornivorous and a bornivore if it absorbs every bounded set.
In a vector bornology, is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology 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.

Bornology of a topological vector space

Every topological vector space, at least on a non discrete valued field gives a bornology on by defining a subset to be bounded, if and only if for all open sets containing zero there exists a with.
If is a locally convex topological vector space then is bounded if and only if all continuous semi-norms on are bounded on.
The set of all bounded subsets of is called the bornology or the Von-Neumann bornology of.

Induced topology

Suppose that we start with a vector space and convex vector bornology on.
If we let denote the collection of all sets that are convex, balanced, and bornivorous then forms neighborhood basis at 0 for a locally convex topology on that is compatible with the vector space structure of.

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.
Note that if is a TVS in which every bornivorous set is a neighborhood of the origin, then any bounded linear map from into any other TVS is continuous.
  1. Every bounded linear operator from into another TVS is continuous.
  2. Every bounded linear operator from into a complete metrizable TVS is continuous.
while if is a Hausdorff locally convex space then we may add to this list:

Examples

The following topological vector spaces are all bornological:
A disk in a topological vector space is called infrabornivorous if it absorbs all Banach disks.
If 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 ultrabornological if any of the following conditions hold: