Vector bornology


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

Definitions

Prerequisites

A on a set is a collection of subsets of that satisfy all the following conditions:
  1. covers that is,
  2. is stable under inclusions; that is, if and then
  3. is stable under finite unions; that is, if then
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 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.

Vector bornology

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.
If is a 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
A vector bornology is called a if it is stable under the formation of convex hulls 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.

Characterizations

Suppose that is a vector space over the field of real or complex numbers and is a bornology on
Then the following are equivalent:
  1. is a vector bornology
  2. addition and scalar multiplication are bounded maps
  3. the balanced hull of every element of is an element of and the sum of any two elements of is again an element of

    Bornology on a topological vector space

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.

Topology induced by a vector 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.

Examples

Locally convex space of bounded functions

Let be the real or complex numbers, 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.

Bornology of equicontinuity

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