Vector bornology


In mathematics, especially functional analysis, a bornology ℬ on a vector space X over a field ?, where ? has a bornology ℬ?, is called a vector bornology if ℬ makes the vector space operations into bounded maps.

Definitions

Prerequisits

A bornology on a set X is a collection ℬ of subsets of X such that

  1. ℬ covers X ;
  2. ℬ is stable under inclusion ;
  3. ℬ is stable under finite unions.
in which case the pair is called a bounded structure.
Elements of ℬ are called ℬ-bounded sets or simply bounded.
A subset ? of a bornology ℬ is called a base or fundamental system of ℬ if for every B ∈ ℬ, there exists an A ∈ ? such that B ⊆ A.
Given a collection ? of subsets of X, the smallest bornology containing ? is called the bornology generated by ?.
If and are bounded structures and f : X → Y is a map then f is called locally bounded or just bounded if the image under f of every ?-bounded set is a ℬ-bounded set;
that is, if for every A ∈ ?, f ∈ ℬ.
If and are bounded structures then the product bornology on X × Y is the bornology having as a base the collection of all sets of the form A × B, where A ∈ ? and B ∈ ℬ.
One may show that a subset of X × Y is bounded in the product bornology if and only if its image under the canonical projections onto X and Y are both bounded.

Vector bornology

Let X be a vector space over a field ?, where ? has a bornology ℬ?.
A bornology ℬ on X is called a vector bornology if its vector spaces operations of addition and scalar multiplication are bounded maps.
Explicitly, this means that when X is endowed with the bornology ℬ then:

  1. the addition map X × X → X defined by ↦ x + y is a bounded map, where X × X has the product bornology, and
  2. the scalar multiplication map ? × X → X defined by ↦ sx is a bounded map, where ? × X has the product bornology induced by and.
Usually, ? is either the real or complex numbers, in which case we call a vector bornology ℬ on X a convex vector bornology if ℬ has a base consisting of convex sets.

Characterizations

Suppose that X is a topological vector space over the field ? of real or complex numbers and ℬ is a bornology on X.
Then the following are equivalent:

  1. ℬ is a vector bornology;
  2. addition and scalar multiplication are bounded maps.
  3. the balanced hull of every element of ℬ is an element of ℬ and the sum of any two elements of ℬ is again an element of ℬ.

Bornology on a topological vector space

If X is a topological vector space then the set of all bounded subsets of X from a vector bornology on X called the von Neumann bornology of X, the usual bornology, or simply the bornology of X and is referred to as natural boundedness.
In any locally convex TVS X, the set of all closed bounded disks form a base for the usual bornology of X.
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 X is a vector space over the field ? of real or complex numbers and ℬ is a vector bornology on X.
Let ? denote all those subsets N of X that are convex, balanced, and bornivorous.
Then ? forms a neighborhood basis at the origin for a locally convex TVS topology.

Examples

Locally convex space of bounded functions

Let ? be the real or complex numbers, let be a bounded structure, and let LB denote the vector space of all locally bounded ?-valued maps on T.
For every B ∈ ℬ, let for all f ∈ LB, where this defines a seminorm on X.
The locally convex TVS topology on LB defined by the family of seminorms is called the topology of uniform convergence on bounded set.
This topology makes LB into a complete space.

Bornology of equicontinuity

Let T be a topological space, ? be the real or complex numbers, and let C denote the vector space of all continuous ?-valued maps on T.
The set â„° of all equicontinuous subsets of C forms a vector bornology on C.