Bornology


In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness.

Definitions

A bornology or boundedness 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 or a bornological set.
Elements of ℬ are called ℬ-bounded sets or simply bounded sets, if ℬ is understood.
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.
A subset ? of a bornology ℬ is called a subbase of ℬ if the collection of all finite unions of sets in ? forms a base for ℬ.
If ? and ℬ are bornologies on X then we say that ℬ is finer or stronger than ? and that ? is coarser or weaker than ℬ if ? ⊆ ℬ.
If is a bounded structure and X ∉ ℬ, then the set of complements is a filter called the filter at infinity.
Given a collection ? of subsets of X, the smallest bornology containing ? is called the bornology generated by ?.
If f : S → X is a map and ℬ is a bornology on X, then we denote the bornology generated by by and called it the inverse image bornology or the initial bornology induced by f on S.

Morphisms: Bounded maps

Suppose that and are bounded structures.
A map f : X → Y 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.
Since the composition of two locally bounded map is again locally bounded, it is clear that the class of all bounded structures forms a category whose morphisms are bounded maps.
An isomorphism in this category is called a bornomorphism and it is a bijective locally bounded map whose inverse is also locally bounded.

Characterizations

Suppose that X and Y are topological vector spaces and f : X → Y is a linear map.
Then the following are equivalent:
if in addition X and Y are locally convex then we may add to this list:

  1. f takes bounded disks to bounded disks;
if in addition X is a seminormed space and Y is locally convex then we may add to this list:

  1. f maps null sequences into bounded subsets of Y.

Examples

If X and Y are any two topological vector spaces and if f : X → Y is a continuous linear operator between them, then f is a bounded linear operator.
The converse is in general false.
A sequentially continuous map f : X → Y between two TVSs is necessarily locally bounded.

Examples and sufficient conditions

;Discrete bornology
For any set X, the power set of X is a bornology on X called the discrete bornology.
;Compact bornology
For any topological space X, the set of all relatively compact subsets of X form a bornology on X called the compact bornology on X.
;Closure and interior bornologies
Suppose that X is a topological space and ℬ is a bornology on X.
The bornology generated by the set of all interiors of sets in ℬ is called the interior of ℬ and is denoted by int ℬ.
The bornology ℬ is called open if ℬ = int ℬ.
Then the bornology generated by the set of all closures of sets in ℬ is called the closure of ℬ and is denoted by cl ℬ.
We necessarily have int ℬ ⊆ ℬ ⊆ cl ℬ.
The bornology ℬ is called closed if it satisfies any of the following equivalent conditions:

  1. ℬ = cl ℬ;
  2. the closed subsets of X generate ℬ;
  3. the closure of every B ∈ ℬ belongs to ℬ.
The bornology ℬ is called proper if ℬ is both open and closed.
The topological space X is called locally ℬ-bounded or just locally bounded if every x ∈ X has a neighborhood that belongs to ℬ.
Every compact subset of a locally bounded topological space is bounded.
;Topological rings
Suppose that X is a commutative topological ring.
A subset S of X is called bounded if for each neighborhood U of 0 in X, there exists a neighborhood V of 0 in X such that SV ⊆ U.

Bornology on a topological vector space

If X is a topological vector space then the set of all boundedness subsets of X from a 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.

Trivial examples

Limits, products, subspaces, quotients

Inverse image bornology

Let S be a set, be an I-indexed family of bounded structures, and let be an I-indexed family of maps where for all i ∈ I, fi : S → Ti.
The inverse image bornology ? on S determined by these maps is the strongest bornology on S making each fi : → locally bounded.
This bornology is equal to.

Direct image bornology

Let S be a set, be an I-indexed family of bounded structures, and let be an I-indexed family of maps where for all i ∈ I, fi : Ti → S.
The direct image bornology ? on S determined by these maps is the weakest bornology on S making each fi : → locally bounded.
If for each i ∈ I, ?i denotes the bornology generated by f, then this bornology is equal to the collection of all subsets A of S of the form where each Ai ∈ ?i and all but finitely many Ai are empty.

Subspace bornology

Suppose that is a bounded structure and S be a subset of X.
The subspace bornology ? on S is the finest bornology on S making the natural inclusion map of S into X, Id : →, locally bounded.

Product bornology

Let be an I-indexed family of bounded structures, let X =, and for each i ∈ I, let fi : X → Xi denote the canonical projection.
The product bornology on X is the inverse image bornology determined by the canonical projections fi : X → Xi.
That is, it is the strongest bornology on X making each of the canonical projections locally bounded.
A base for the product bornology is given by