View source for Bornology

{{Short description|Mathematical generalization of boundedness}}
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 [[Bounded set (functional analysis)|boundedness]]. One of the key motivations behind bornologies and bornological analysis is the fact that [[bornological space]]s provide a convenient setting for [[homological algebra]] in functional analysis. This is because<ref>{{cite arXiv|last1=Block|first1=Jonathan|last2=Daenzer|first2=Calder|date=2009-01-09|title=Mukai duality for gerbes with connection|class=math.QA|eprint=0803.1529}}</ref><sup>pg 9</sup> the [[category (mathematics)|category]] of bornological spaces is [[additive category|additive]], [[Complete category|complete]], [[cocomplete]], and has a tensor product [[Adjoint functors|adjoint]] to an [[internal hom]], all necessary components for homological algebra.

==History==

Bornology originates from [[functional analysis]]. There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies ([[Vector topology|vector topologies]], [[Continuous linear operator|continuous operators]], [[Open set|open]]/[[Compact space|compact]] subsets, etc.) and the other is to study notions related to boundedness{{sfn|Hogbe-Nlend|1971|p=5}} ([[Vector bornology|vector bornologies]], [[bounded operator]]s, [[Bounded set (topological vector space)|bounded subsets]], etc.). 

For [[normed space]]s, from which functional analysis arose, topological and bornological notions are distinct but complementary and closely related. 
For example, the unit ball centered at the origin is both a [[Neighbourhood (mathematics)|neighborhood of the origin]] and a bounded subset. 
Furthermore, a subset of a normed space is ''a neighborhood of the origin'' (respectively, is ''a bounded set'') exactly when it ''contains'' (respectively, it ''is contained in'') a non-zero scalar multiple of this ball; so this is one instance where the topological and bornological notions are distinct but complementary (in the sense that their definitions differ only by which of <math>\,\subseteq\,</math> and <math>\,\supseteq\,</math> is used).
Other times, the distinction between topological and bornological notions may even be unnecessary. 
For example, for linear maps between normed spaces, being [[Continuous linear operator|continuous]] (a topological notion) is equivalent to being [[Bounded operator|bounded]] (a bornological notion). 
Although the distinction between topology and bornology is often blurred or unnecessary for normed space, it becomes more important when studying generalizations of normed spaces. 
Nevertheless, bornology and topology can still be thought of as two necessary, distinct, and complementary aspects of one and the same reality.{{sfn|Hogbe-Nlend|1971|p=5}} 

The general theory of [[topological vector space]]s arose first from the theory of normed spaces and then bornology emerged from this general theory of topological vector spaces, although bornology has since become recognized as a fundamental notion in [[functional analysis]].{{sfn|Hogbe-Nlend|1971|pp=1-2}} 
Born from the work of [[George Mackey]] (after whom [[Mackey space]]s are named), the importance of bounded subsets first became apparent in [[Duality theory (functional analysis)|duality theory]], especially because of the [[Mackey–Arens theorem]] and the [[Mackey topology]].{{sfn|Hogbe-Nlend|1971|pp=1-2}} 
Starting around the 1950s, it became apparent that topological vector spaces were inadequate for the study of certain major problems.{{sfn|Hogbe-Nlend|1971|pp=1-2}} 
For example, the multiplication operation of some important [[topological algebra]]s was not continuous, although it was often bounded.{{sfn|Hogbe-Nlend|1971|pp=1-2}} 
Other major problems for which TVSs were found to be inadequate was in developing a more general theory of differential calculus, generalizing [[Distribution (mathematics)|distributions]] from (the usual) scalar-valued distributions to vector or operator-valued distributions, and extending the holomorphic [[functional calculus]] of [[Gelfand representation|Gelfand]] (which is primarily concerted with [[Banach algebra]]s or [[locally convex algebra]]s) to a broader class of operators, including those whose [[Spectrum (functional analysis)|spectra]] are not compact. 
Bornology has been found to be a useful tool for investigating these problems and others,{{sfn|Hogbe-Nlend|1971}} including problems in [[algebraic geometry]] and [[general topology]].

==Definitions==

A {{em|bornology}} on a set is a [[Cover (topology)|cover]] of the set that is closed under finite unions and taking subsets. Elements of a bornology are called {{em|bounded sets}}.

Explicitly, a {{em|{{visible anchor|bornology}}}} or {{em|{{visible anchor|boundedness}}}} on a set <math>X</math> is a [[Family of sets|family]] <math>\mathcal{B} \neq \varnothing</math> of subsets of <math>X</math> such that
<ol>
<li><math>\mathcal{B}</math> is ''stable under inclusion'' or {{em|[[Downward closed|{{visible anchor|downward closed}}]]}}: If <math>B \in \mathcal{B}</math> then every subset of <math>B</math> is an element of <math>\mathcal{B}.</math>
* Stated in [[plain English]], this says that subsets of bounded sets are bounded.</li>
<li><math>\mathcal{B}</math> ''[[Cover (topology)|covers]]'' <math>X:</math> Every point of <math>X</math> is an element of some <math>B \in \mathcal{B},</math> or equivalently, <math>X = {\textstyle\bigcup\limits_{B \in \mathcal{B}} B}.</math>
* Assuming (1), this condition may be replaced with: For every <math>x \in X,</math> <math>\{x\} \in \mathcal{B}.</math> In plain English, this says that every point is bounded.</li>
<li><math>\mathcal{B}</math> is ''stable under finite unions'': The [[Union (set theory)|union]] of finitely many elements of <math>\mathcal{B}</math> is an element of <math>\mathcal{B},</math> or equivalently, the union of any {{em|two}} sets belonging to <math>\mathcal{B}</math> also belongs to <math>\mathcal{B}.</math>
* In plain English, this says that the union of two bounded sets is a bounded set.</li>
</ol>
in which case the pair <math>(X, \mathcal{B})</math> is called a {{em|{{visible anchor|bounded structure}}}} or a {{em|{{visible anchor|bornological set}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}} 

Thus a bornology can equivalently be defined as a [[#downward closed|downward closed]] cover that is closed under [[Union (set theory)|binary unions]].
A non-empty family of sets that closed under finite unions and taking subsets (properties (1) and (3)) is called an {{em|[[Ideal (set theory)|ideal]]}} (because it is an [[Ideal (order theory)|ideal]] in the [[Boolean algebra]]/[[field of sets]] consisting of [[Power set|all subsets]]). A bornology on a set <math>X</math> can thus be equivalently defined as an ideal that covers <math>X.</math> 

Elements of <math>\mathcal{B}</math> are called {{em|<math>\mathcal{B}</math>-bounded sets}} or simply {{em|{{visible anchor|bounded set}}s}}, if <math>\mathcal{B}</math> is understood. 
Properties (1) and (2) imply that every singleton subset of <math>X</math> is an element of every bornology on <math>X;</math> property (3), in turn, guarantees that the same is true of every finite subset of <math>X.</math> In other words, points and finite subsets are always bounded in every bornology. In particular, the empty set is always bounded. 

If <math>(X, \mathcal{B})</math> is a bounded structure and <math>X \notin \mathcal{B},</math> then the set of complements <math>\{X \setminus B : B \in \mathcal{B}\}</math> is a (proper) [[Filter (set theory)|filter]] called the {{em|{{visible anchor|filter at infinity}}}};{{sfn|Narici|Beckenstein|2011|pp=156-175}} it is always a {{em|[[free filter]]}}, which by definition means that it has empty intersection/[[Kernel (set theory)|kernel]], because <math>\{x\} \in \mathcal{B}</math> for every <math>x \in X.</math> 

===Bases and subbases===

If <math>\mathcal{A}</math> and <math>\mathcal{B}</math> are bornologies on <math>X</math> then <math>\mathcal{B}</math> is said to be {{em|{{visible anchor|finer}}}} or {{em|{{visible anchor|stronger}}}} than <math>\mathcal{A}</math> and also <math>\mathcal{A}</math> is said to be {{em|{{visible anchor|coarser}}}} or {{em|{{visible anchor|weaker}}}} than <math>\mathcal{B}</math> if <math>\mathcal{A} \subseteq \mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 

A [[family of sets]] <math>\mathcal{A}</math> is called a {{em|{{visible anchor|base for a bornology|base|text=base}}}} or {{em|{{visible anchor|fundamental system}}}} of a bornology <math>\mathcal{B}</math> if <math>\mathcal{A} \subseteq \mathcal{B}</math> and for every <math>B \in \mathcal{B},</math> there exists an <math>A \in \mathcal{A}</math> such that <math>B \subseteq A.</math> 

A family of sets <math>\mathcal{S}</math> is called a {{em|{{visible anchor|subbase for a bornology|subbase|text=subbase}}}} of a bornology <math>\mathcal{B}</math> if <math>\mathcal{S} \subseteq \mathcal{B}</math> and the collection of all finite unions of sets in <math>\mathcal{S}</math> forms a base for <math>\mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 

Every base for a bornology is also a subbase for it.

===Generated bornology===

The intersection of any collection of (one or more) bornologies on <math>X</math> is once again a bornology on <math>X.</math> 
Such an intersection of bornologies will cover <math>X</math> because every bornology on <math>X</math> contains every finite subset of <math>X</math> (that is, if <math>\mathcal{B}</math> is a bornology on <math>X</math> and <math>F \subseteq X</math> is finite then <math>F \in \mathcal{B}</math>). It is readily verified that such an intersection will also be closed under (subset) inclusion and finite unions and thus will be a bornology on <math>X.</math> 

Given a collection <math>\mathcal{S}</math> of subsets of <math>X,</math> the smallest bornology on <math>X</math> containing <math>\mathcal{S}</math> is called the {{em|{{visible anchor|generated bornology|text=bornology generated by <math>\mathcal{S}</math>}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
It is equal to the intersection of all bornologies on <math>X</math> that contain <math>\mathcal{S}</math> as a subset. 
This intersection is well-defined because the [[power set]] <math>\wp(X)</math> of <math>X</math> is always a bornology on <math>X,</math> so every family <math>\mathcal{S}</math> of subsets of <math>X</math> is always contained in at least one bornology on <math>X.</math>

==Bounded maps==

Suppose that <math>(X, \mathcal{A})</math> and <math>(Y, \mathcal{B})</math> are bounded structures. 
A map <math>f : X \to Y</math> is called a {{em|{{visible anchor|locally bounded map}}}}, or just a {{em|{{visible anchor|bounded map}}}}, if the image under <math>f</math> of every <math>\mathcal{A}</math>-bounded set is a <math>\mathcal{B}</math>-bounded set; 
that is, if for every <math>A \in \mathcal{A},</math> <math>f(A) \in \mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}

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 (mathematics)|category]] whose [[morphisms]] are bounded maps. 
An [[isomorphism]] in this category is called a {{em|{{visible anchor|bornomorphism}}}} and it is a [[bijective]] locally bounded map whose inverse is also locally bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

===Examples of bounded maps===

If <math>f : X \to Y</math> is a continuous linear operator between two topological vector spaces (not necessarily Hausdorff), then it is a bounded linear operator when <math>X</math> and <math>Y</math> have their [[Bornological space#Bornology of a topological vector space|von-Neumann bornologies]], where a set is bounded precisely when it is absorbed by all neighbourhoods of origin (these are the subsets of a TVS that are normally called bounded when no other bornology is explicitly mentioned.). 
The converse is in general false.

A [[sequentially continuous]] map <math>f : X \to Y</math> between two TVSs is necessarily locally bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

==General constructions==

'''Discrete bornology'''

For any set <math>X,</math> the [[power set]] <math>\wp(X)</math> of <math>X</math> is a bornology on <math>X</math> called the {{em|{{visible anchor|discrete bornology}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}} Since every bornology on <math>X</math> is a subset of <math>\wp(X),</math> the discrete bornology is the finest bornology on <math>X.</math> 
If <math>(X, \mathcal{B})</math> is a bounded structure then (because bornologies are [[#downward closed|downward closed]]) <math>\mathcal{B}</math> is the discrete bornology if and only if <math>X \in \mathcal{B}.</math>

'''Indiscrete bornology'''

For any set <math>X,</math> the set of all finite subsets of <math>X</math> is a bornology on <math>X</math> called the {{em|{{visible anchor|indiscrete bornology}}}}. It is the coarsest bornology on <math>X,</math> meaning that it is a subset of every bornology on <math>X.</math> 

'''Sets of bounded cardinality'''

The set of all [[Countable set|countable]] subsets of <math>X</math> is a bornology on <math>X.</math> 
More generally, for any infinite [[Cardinal number|cardinal]] <math>\kappa,</math> the set of all subsets of <math>X</math> having cardinality at most <math>\kappa</math> is a bornology on <math>X.</math>

===Inverse image bornology===

If <math>f : S \to X</math> is a map and <math>\mathcal{B}</math> is a bornology on <math>X,</math> then <math>\left[f^{-1}(\mathcal{B})\right]</math> denotes the bornology generated by <math>f^{-1}(\mathcal{B}) := \left\{f^{-1}(B) : B \in \mathcal{B}\right\},</math> which is called it the {{em|inverse image bornology}} or the {{em|initial bornology}} induced by <math>f</math> on <math>S.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}

Let <math>S</math> be a set, <math>\left(T_i, \mathcal{B}_i\right)_{i \in I}</math> be an <math>I</math>-indexed family of bounded structures, and let <math>\left(f_i\right)_{i \in I}</math> be an <math>I</math>-indexed family of maps where <math>f_i : S \to T_i</math> for every <math>i \in I.</math> 
The {{em|{{visible anchor|inverse image bornology}}}} <math>\mathcal{A}</math> on <math>S</math> determined by these maps is the strongest bornology on <math>S</math> making each <math>f_i : (S, \mathcal{A}) \to \left(T_i, \mathcal{B}_i\right)</math> locally bounded. 
This bornology is equal to{{sfn|Narici|Beckenstein|2011|pp=156-175}} <math>{\textstyle \bigcap\limits_{i \in I} \left[f^{-1}\left(\mathcal{B}_i\right)\right]}.</math>

===Direct image bornology===

Let <math>S</math> be a set, <math>\left(T_i, \mathcal{B}_i\right)_{i \in I}</math> be an <math>I</math>-indexed family of bounded structures, and let <math>\left(f_i\right)_{i \in I}</math> be an <math>I</math>-indexed family of maps where <math>f_i : T_i \to S</math> for every <math>i \in I.</math> 
The {{em|{{visible anchor|direct image bornology}}}} <math>\mathcal{A}</math> on <math>S</math> determined by these maps is the weakest bornology on <math>S</math> making each <math>f_i : \left(T_i, \mathcal{B}_i\right) \to (S, \mathcal{A})</math> locally bounded. 
If for each <math>i \in I,</math> <math>\mathcal{A}_i</math> denotes the bornology generated by <math>f\left(\mathcal{B}_i\right),</math> then this bornology is equal to the collection of all subsets <math>A</math> of <math>S</math> of the form <math>\cup_{i \in I} A_i</math> where each <math>A_i \in \mathcal{A}_i</math> and all but finitely many <math>A_i</math> are empty.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

===Subspace bornology===

Suppose that <math>(X, \mathcal{B})</math> is a bounded structure and <math>S</math> be a subset of <math>X.</math> 
The {{em|{{visible anchor|subspace bornology}}}} <math>\mathcal{A}</math> on <math>S</math> is the finest bornology on <math>S</math> making the [[inclusion map]] <math>(S, \mathcal{A}) \to (X, \mathcal{B})</math> of <math>S</math> into <math>X</math> (defined by <math>s \mapsto s</math>) locally bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

===Product bornology===

Let <math>\left(X_i, \mathcal{B}_i\right)_{i \in I}</math> be an <math>I</math>-indexed family of bounded structures, let <math>X = {\textstyle \prod\limits_{i \in I} X_i},</math> and for each <math>i \in I,</math> let <math>f_i : X \to X_i</math> denote the canonical projection. 
The {{em|{{visible anchor|product bornology}}}} on <math>X</math> is the inverse image bornology determined by the canonical projections <math>f_i : X \to X_i.</math> 
That is, it is the strongest bornology on <math>X</math> making each of the canonical projections locally bounded. 
A base for the product bornology is given by <math>{\textstyle \left\{\prod\limits_{i \in I} B_i ~:~ B_i \in \mathcal{B}_i \text{ for all } i \in I\right\}}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}

==Topological constructions==


===Compact bornology===

A subset of a [[topological space]] <math>X</math> is called [[relatively compact subspace|relatively compact]] if its closure is a [[Compact space|compact subspace]] of <math>X.</math> 
For any topological space <math>X</math> in which singleton subsets are relatively compact (such as a [[T1 space]]), the set of all relatively compact subsets of <math>X</math> form a bornology on <math>X</math> called the {{em|{{visible anchor|compact bornology}}}} on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
Every continuous map between [[T1 space]]s is bounded with respect to their compact bornologies. 

The set of [[relatively compact subspace|relatively compact]] subsets of <math>\R</math> form a bornology on <math>\R.</math> A base for this bornology is given by all closed intervals of the form <math>[-n, n]</math> for <math>n = 1, 2, 3, \ldots.</math>

===Metric bornology===

Given a [[metric space]] <math>(X, d),</math> the {{em|metric bornology}} consists of all subsets <math>S \subseteq X</math> such that the [[supremum]] <math>\sup_{s, t \in S} d(s, t) < \infty</math> is finite. 

Similarly, given a [[measure space]] <math>(X, \Omega, \mu),</math> the family of all measurable subsets <math>S \in \Omega</math> of finite measure (meaning <math>\mu(S) < \infty</math>) form a bornology on <math>X.</math> 

===Closure and interior bornologies===

Suppose that <math>X</math> is a [[topological space]] and <math>\mathcal{B}</math> is a bornology on <math>X.</math> 

The bornology generated by the set of all [[topological interior]]s of sets in <math>\mathcal{B}</math> (that is, generated by <math>\{\operatorname{int} B : B \in \mathcal{B}\}</math> is called the {{em|{{visible anchor|interior bornology|text=interior}}}} of <math>\mathcal{B}</math> and is denoted by <math>\operatorname{int} \mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
The bornology <math>\mathcal{B}</math> is called {{em|{{visible anchor|open bornology|text=open}}}} if <math>\mathcal{B} = \operatorname{int} \mathcal{B}.</math> 

The bornology generated by the set of all [[topological closure]]s of sets in <math>\mathcal{B}</math> (that is, generated by <math>\{\operatorname{cl} B : B \in \mathcal{B}\}</math>) is called the {{em|{{visible anchor|closure bornology|text=closure}}}} of <math>\mathcal{B}</math> and is denoted by <math>\operatorname{cl} \mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
We necessarily have <math>\operatorname{int} \mathcal{B} \subseteq \mathcal{B} \subseteq \operatorname{cl} \mathcal{B}.</math>

The bornology <math>\mathcal{B}</math> is called {{em|{{visible anchor|closed bornology|text=closed}}}} if it satisfies any of the following equivalent conditions: 
<ol>
<li><math>\mathcal{B} = \operatorname{cl} \mathcal{B};</math></li>
<li>the closed subsets of <math>X</math> generate <math>\mathcal{B}</math>;{{sfn|Narici|Beckenstein|2011|pp=156-175}} </li>
<li>the closure of every <math>B \in \mathcal{B}</math> belongs to <math>\mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} </li>
</ol>
The bornology <math>\mathcal{B}</math> is called {{em|{{visible anchor|proper bornology|text=proper}}}} if <math>\mathcal{B}</math> is both open and closed.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

The topological space <math>X</math> is called {{em|{{visible anchor|locally bounded topological space|text=locally <math>\mathcal{B}</math>-bounded}}}} or just {{em|locally bounded}} if every <math>x \in X</math> has a neighborhood that belongs to <math>\mathcal{B}.</math>
Every compact subset of a locally bounded topological space is bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

===Bornology of a topological vector space===

{{See also|Vector bornology}}

If <math>X</math> is a [[topological vector space]] (TVS) then the set of all [[Bounded set (functional analysis)|bounded]] subsets of <math>X</math> form a bornology (indeed, even a [[vector bornology]]) on <math>X</math> called the {{em|{{visible anchor|von Neumann bornology|text=[[John von Neumann|von Neumann]] bornology of <math>X</math>}}}}, the {{em|{{visible anchor|usual bornology}}}}, or simply {{em|{{visible anchor|the bornology}}}} of <math>X</math> and is referred to as {{em|{{visible anchor|natural boundedness}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
In any [[Locally convex topological vector space|locally convex]] TVS <math>X,</math> the set of all closed bounded [[Absolutely convex set|disks]] forms a base for the usual bornology of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 

A linear map between two [[bornological space]]s is [[Continuous linear operator|continuous]] if and only if it is bounded (with respect to the usual bornologies).

===Topological rings===

Suppose that <math>X</math> is a commutative [[topological ring]]. 
A subset <math>S</math> of <math>X</math> is called a {{em|{{visible anchor|bounded subset of a topological ring|text=bounded set}}}} if for each neighborhood <math>U</math> of the origin in <math>X,</math> there exists a neighborhood <math>V</math> of the origin in <math>X</math> such that <math>S V \subseteq U.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}

==See also==

* {{annotated link|Bornivorous set}}
* {{annotated link|Bornological space}}
* {{annotated link|Coarse structure#bounded set}}
* {{annotated link|Space of linear maps}}
* {{annotated link|Ultrabornological space}}
* {{annotated link|Vector bornology}}

==References==

{{reflist}}

* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!--{{sfn|Hogbe-Nlend|1977|p=}}-->
* {{cite journal|last=Hogbe-Nlend|first=Henri|author-link=Henri Hogbe-Nlend|title=Les racines historiques de la bornologie moderne|journal=Séminaire Choquet: Initiation à l'analyse|language=French|volume=10|issue=1|date=1971|pages=1–7|doi=|mr=477660|url=http://www.numdam.org/item/SC_1970-1971__10_1_A5_0.pdf}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces|pp=29–33, 49, 104}} <!--{{sfn|Khaleelulla|1982|p=}}-->
* {{Kriegl Michor The Convenient Setting of Global Analysis}} <!--{{sfn|Kriegl|Michor|1997|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->

{{Boundedness and bornology}}
{{Topological vector spaces}}

[[Category:Functional analysis]]
[[Category:Topological vector spaces]]