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{{more footnotes|date=June 2020}}

In [[mathematics]], especially [[functional analysis]], a [[bornology]] <math>\mathcal{B}</math> on a [[vector space]] <math>X</math> over a [[Field (mathematics)|field]] <math>\mathbb{K},</math> where <math>\mathbb{K}</math> has a bornology ℬ<sub><math>\mathbb{F}</math></sub>, is called a '''vector bornology''' if <math>\mathcal{B}</math> makes the vector space operations into bounded maps.

== Definitions ==
=== Prerequisits ===
{{Main|Bornology}}

A {{em|'''bornology'''}} on a set <math>X</math> is a collection <math>\mathcal{B}</math> of subsets of <math>X</math> that satisfy all the following conditions:
#<math>\mathcal{B}</math> covers <math>X;</math> that is, <math>X = \cup \mathcal{B}</math>
#<math>\mathcal{B}</math> is stable under inclusions; that is, if <math>B \in \mathcal{B}</math> and <math>A \subseteq B,</math> then <math>A \in \mathcal{B}</math>
#<math>\mathcal{B}</math> is stable under finite unions; that is,  if <math>B_1, \ldots, B_n \in \mathcal{B}</math> then <math>B_1 \cup \cdots \cup B_n \in \mathcal{B}</math>

Elements of the collection <math>\mathcal{B}</math> are called {{em|'''<math>\mathcal{B}</math>-bounded'''}} or simply {{em|'''bounded sets'''}} if <math>\mathcal{B}</math> is understood. 
The pair <math>(X, \mathcal{B})</math> is called a {{em|'''bounded structure'''}} or a {{em|'''bornological set'''}}.

A {{em|'''base'''}} or {{em|'''fundamental system'''}} of a bornology <math>\mathcal{B}</math> is a subset <math>\mathcal{B}_0</math> of <math>\mathcal{B}</math> such that each element of <math>\mathcal{B}</math> is a subset of some element of <math>\mathcal{B}_0.</math> Given a collection <math>\mathcal{S}</math> of subsets of <math>X,</math> the smallest bornology containing <math>\mathcal{S}</math> is called the '''bornology generated by <math>\mathcal{S}.</math>'''{{sfn|Narici|Beckenstein|2011|pp=156–175}}

If <math>(X, \mathcal{B})</math> and <math>(Y, \mathcal{C})</math> are bornological sets then their {{em|'''product bornology'''}} on <math>X \times Y</math> is the bornology having as a base the collection of all sets of the form <math>B \times C,</math> where <math>B \in \mathcal{B}</math> and <math>C \in \mathcal{C}.</math>{{sfn|Narici|Beckenstein|2011|pp=156–175}} 
A subset of <math>X \times Y</math> is bounded in the product bornology if and only if its image under the canonical projections onto <math>X</math> and <math>Y</math> are both bounded. 

If <math>(X, \mathcal{B})</math> and <math>(Y, \mathcal{C})</math> are bornological sets then a function <math>f : X \to Y</math> is said to be a {{em|'''locally bounded map'''}} or a {{em|'''bounded map'''}} (with respect to these bornologies) if it maps <math>\mathcal{B}</math>-bounded subsets of <math>X</math> to <math>\mathcal{C}</math>-bounded subsets of <math>Y;</math> that is, if <math>f\left(\mathcal{B}\right) \subseteq \mathcal{C}.</math>{{sfn|Narici|Beckenstein|2011|pp=156–175}} 
If in addition <math>f</math> is a bijection and <math>f^{-1}</math> is also bounded then <math>f</math> is called a {{em|'''bornological isomorphism'''}}. 

=== Vector bornology ===

Let <math>X</math> be a vector space over a [[Field (mathematics)|field]] <math>\mathbb{K}</math> where <math>\mathbb{K}</math> has a bornology <math>\mathcal{B}_{\mathbb{K}}.</math> 
A bornology <math>\mathcal{B}</math> on <math>X</math> is called a {{em|'''vector bornology on <math>X</math>'''}} if it is stable under vector addition, scalar multiplication, and the formation of [[balanced hull]]s (i.e. if the sum of two bounded sets is bounded, etc.).

If <math>X</math> is a vector space and <math>\mathcal{B}</math> is a bornology on <math>X,</math> then the following are equivalent: 
#<math>\mathcal{B}</math> is a vector bornology
#Finite sums and balanced hulls of <math>\mathcal{B}</math>-bounded sets are <math>\mathcal{B}</math>-bounded{{sfn|Narici|Beckenstein|2011|pp=156-175}}
#The scalar multiplication map <math>\mathbb{K} \times X \to X</math> defined by <math>(s, x) \mapsto sx</math> and the addition map <math>X \times X \to X</math> defined by <math>(x, y) \mapsto x + y,</math> are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets){{sfn|Narici|Beckenstein|2011|pp=156-175}}

A vector bornology <math>\mathcal{B}</math> is called a {{em|'''convex vector bornology'''}} if it is stable under the formation of [[convex hull]]s (i.e. the convex hull of a bounded set is bounded) then <math>\mathcal{B}.</math> 
And a vector bornology <math>\mathcal{B}</math> is called {{em|'''separated'''}} if the only bounded vector subspace of <math>X</math> is the 0-dimensional trivial space <math>\{ 0 \}.</math> 

Usually, <math>\mathbb{K}</math> is either the real or complex numbers, in which case a vector bornology <math>\mathcal{B}</math> on <math>X</math> will be called a {{em|'''convex vector bornology'''}} if <math>\mathcal{B}</math> has a base consisting of [[Convex set|convex]] sets.

== Characterizations ==
Suppose that <math>X</math> is a [[vector space]] over the field <math>\mathbb{F}</math> of real or complex numbers and <math>\mathcal{B}</math> is a bornology on <math>X.</math>
Then the following are equivalent:
#<math>\mathcal{B}</math> is a vector bornology
#addition and scalar multiplication are bounded maps{{sfn|Narici|Beckenstein|2011|pp=156–175}}
#the [[Balanced set|balanced hull]] of every element of <math>\mathcal{B}</math> is an element of <math>\mathcal{B}</math> and the sum of any two elements of <math>\mathcal{B}</math> is again an element of <math>\mathcal{B}</math>{{sfn|Narici|Beckenstein|2011|pp=156–175}}

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

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 <math>X</math> is a vector space over the field <math>\mathbb{K}</math> of real or complex numbers and <math>\mathcal{B}</math> is a vector bornology on <math>X.</math>
Let <math>\mathcal{N}</math> denote all those subsets <math>N</math> of <math>X</math> that are convex, [[Balanced set|balanced]], and [[Bornivorous set|bornivorous]].
Then <math>\mathcal{N}</math> forms a neighborhood basis at the origin for a [[locally convex topological vector space]] topology.

== Examples ==
=== Locally convex space of bounded functions ===
Let <math>\mathbb{K}</math> be the real or complex numbers (endowed with their usual bornologies), let <math>(T, \mathcal{B})</math> be a bounded structure, and let <math>LB(T, \mathbb{K})</math> denote the vector space of all locally bounded <math>\mathbb{K}</math>-valued maps on <math>T.</math>
For every <math>B \in \mathcal{B},</math> let <math>p_{B}(f) := \sup \left| f(B) \right|</math> for all <math>f \in LB(T, \mathbb{K}),</math> where this defines a [[seminorm]] on <math>X.</math>
The [[locally convex]] topological vector space topology on <math>LB(T, \mathbb{K})</math> defined by the family of seminorms <math>\left\{ p_{B} : B \in \mathcal{B} \right\}</math> is called the {{em|'''topology of uniform convergence on bounded set'''}}.{{sfn|Narici|Beckenstein|2011|pp=156–175}}
This topology makes <math>LB(T, \mathbb{K})</math> into a [[complete space]].{{sfn|Narici|Beckenstein|2011|pp=156–175}}

=== Bornology of equicontinuity ===
Let <math>T</math> be a topological space, <math>\mathbb{K}</math> be the real or complex numbers, and let <math>C(T, \mathbb{K})</math> denote the vector space of all continuous <math>\mathbb{K}</math>-valued maps on <math>T.</math>
The set of all [[equicontinuous]] subsets of <math>C(T, \mathbb{K})</math> forms a vector bornology on <math>C(T, \mathbb{K}).</math>{{sfn|Narici|Beckenstein|2011|pp=156–175}}

== See also ==
* [[Bornivorous set]]
* [[Bornological space]]
* [[Bornology]]
* [[Space of linear maps]]
* [[Ultrabornological space]]

==Citations==
{{Reflist|30em}}

==Bibliography==
{{refbegin}}
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn|Hogbe-Nlend|1977|p=}} -->
* {{Cite book| title = The Convenient Setting of Global Analysis
 | last1 = Kriegl | first1 = Andreas
 | last2 = Michor | first2 = Peter W.
 | year = 1997
 | publisher = [[American Mathematical Society]]
 | series = Mathematical Surveys and Monographs
 | isbn = 978-082180780-4
}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
{{refend}}

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

[[Category:Topological vector spaces]]