bornology

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 topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness{{sfn|Hogbe-Nlend|1971|p=5}} (vector bornologies, bounded operators, bounded subsets, etc.).

For normed spaces, 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 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 $\backslash ,\backslash subseteq\backslash ,$ and $\backslash ,\backslash supseteq\backslash ,$ 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 (a topological notion) is equivalent to being 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 spaces 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 spaces are named), the importance of bounded subsets first became apparent in 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 algebras 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 distributions from (the usual) scalar-valued distributions to vector or operator-valued distributions, and extending the holomorphic functional calculus of Gelfand (which is primarily concerted with Banach algebras or locally convex algebras) to a broader class of operators, including those whose 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.

A {{em|bornology}} on a set is a 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 $X$ is a family $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ of subsets of $X$ such that

1. $\backslash mathcal\{B\}$ is stable under inclusion or {{em|Downward closed}}: If $B\; \backslash in\; \backslash mathcal\{B\}$ then every subset of $B$ is an element of $\backslash mathcal\{B\}.$
   • Stated in plain English, this says that subsets of bounded sets are bounded.

2. $\backslash mathcal\{B\}$ covers $X:$ Every point of $X$ is an element of some $B\; \backslash in\; \backslash mathcal\{B\},$ or equivalently, $X\; =\; \{\backslash textstyle\backslash bigcup\backslash limits\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\; B\}.$
   • Assuming (1), this condition may be replaced with: For every $x\; \backslash in\; X,$ $\backslash \{x\backslash \}\; \backslash in\; \backslash mathcal\{B\}.$ In plain English, this says that every point is bounded.

3. $\backslash mathcal\{B\}$ is stable under finite unions: The union of finitely many elements of $\backslash mathcal\{B\}$ is an element of $\backslash mathcal\{B\},$ or equivalently, the union of any {{em|two}} sets belonging to $\backslash mathcal\{B\}$ also belongs to $\backslash mathcal\{B\}.$
   • In plain English, this says that the union of two bounded sets is a bounded set.

in which case the pair $(X,\; \backslash mathcal\{B\})$ 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 cover that is closed under 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}} (because it is an ideal in the Boolean algebra/field of sets consisting of all subsets). A bornology on a set $X$ can thus be equivalently defined as an ideal that covers $X.$

Elements of $\backslash mathcal\{B\}$ are called {{em|$\backslash mathcal\{B\}$-bounded sets}} or simply {{em|{{visible anchor|bounded set}}s}}, if $\backslash mathcal\{B\}$ is understood.

Properties (1) and (2) imply that every singleton subset of $X$ is an element of every bornology on $X;$ property (3), in turn, guarantees that the same is true of every finite subset of $X.$ In other words, points and finite subsets are always bounded in every bornology. In particular, the empty set is always bounded.

If $(X,\; \backslash mathcal\{B\})$ is a bounded structure and $X\; \backslash notin\; \backslash mathcal\{B\},$ then the set of complements $\backslash \{X\; \backslash setminus\; B\; :\; B\; \backslash in\; \backslash mathcal\{B\}\backslash \}$ is a (proper) 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, because $\backslash \{x\backslash \}\; \backslash in\; \backslash mathcal\{B\}$ for every $x\; \backslash in\; X.$

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

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

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

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

The intersection of any collection of (one or more) bornologies on $X$ is once again a bornology on $X.$

Such an intersection of bornologies will cover $X$ because every bornology on $X$ contains every finite subset of $X$ (that is, if $\backslash mathcal\{B\}$ is a bornology on $X$ and $F\; \backslash subseteq\; X$ is finite then $F\; \backslash in\; \backslash mathcal\{B\}$). 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 $X.$

Given a collection $\backslash mathcal\{S\}$ of subsets of $X,$ the smallest bornology on $X$ containing $\backslash mathcal\{S\}$ is called the {{em|{{visible anchor|generated bornology|text=bornology generated by $\backslash mathcal\{S\}$}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

It is equal to the intersection of all bornologies on $X$ that contain $\backslash mathcal\{S\}$ as a subset.

This intersection is well-defined because the power set $\backslash wp(X)$ of $X$ is always a bornology on $X,$ so every family $\backslash mathcal\{S\}$ of subsets of $X$ is always contained in at least one bornology on $X.$

Suppose that $(X,\; \backslash mathcal\{A\})$ and $(Y,\; \backslash mathcal\{B\})$ are bounded structures.

A map $f\; :\; X\; \backslash to\; Y$ is called a {{em|{{visible anchor|locally bounded map}}}}, or just a {{em|{{visible anchor|bounded map}}}}, if the image under $f$ of every $\backslash mathcal\{A\}$-bounded set is a $\backslash mathcal\{B\}$-bounded set;

that is, if for every $A\; \backslash in\; \backslash mathcal\{A\},$ $f(A)\; \backslash in\; \backslash mathcal\{B\}.${{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 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}}

If $f\; :\; X\; \backslash to\; Y$ is a continuous linear operator between two topological vector spaces (not necessarily Hausdorff), then it is a bounded linear operator when $X$ and $Y$ have their 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 $f\; :\; X\; \backslash to\; Y$ between two TVSs is necessarily locally bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

Discrete bornology

For any set $X,$ the power set $\backslash wp(X)$ of $X$ is a bornology on $X$ called the {{em|{{visible anchor|discrete bornology}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}} Since every bornology on $X$ is a subset of $\backslash wp(X),$ the discrete bornology is the finest bornology on $X.$

If $(X,\; \backslash mathcal\{B\})$ is a bounded structure then (because bornologies are downward closed) $\backslash mathcal\{B\}$ is the discrete bornology if and only if $X\; \backslash in\; \backslash mathcal\{B\}.$

Indiscrete bornology

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

Sets of bounded cardinality

The set of all countable subsets of $X$ is a bornology on $X.$

More generally, for any infinite cardinal $\backslash kappa,$ the set of all subsets of $X$ having cardinality at most $\backslash kappa$ is a bornology on $X.$

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

Let $S$ be a set, $\backslash left(T\_i,\; \backslash mathcal\{B\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ be an $I$-indexed family of bounded structures, and let $\backslash left(f\_i\backslash right)\_\{i\; \backslash in\; I\}$ be an $I$-indexed family of maps where $f\_i\; :\; S\; \backslash to\; T\_i$ for every $i\; \backslash in\; I.$

The {{em|{{visible anchor|inverse image bornology}}}} $\backslash mathcal\{A\}$ on $S$ determined by these maps is the strongest bornology on $S$ making each $f\_i\; :\; (S,\; \backslash mathcal\{A\})\; \backslash to\; \backslash left(T\_i,\; \backslash mathcal\{B\}\_i\backslash right)$ locally bounded.

This bornology is equal to{{sfn|Narici|Beckenstein|2011|pp=156-175}} $\{\backslash textstyle\; \backslash bigcap\backslash limits\_\{i\; \backslash in\; I\}\; \backslash left[f^\{-1\}\backslash left(\backslash mathcal\{B\}\_i\backslash right)\backslash right]\}.$

Let $S$ be a set, $\backslash left(T\_i,\; \backslash mathcal\{B\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ be an $I$-indexed family of bounded structures, and let $\backslash left(f\_i\backslash right)\_\{i\; \backslash in\; I\}$ be an $I$-indexed family of maps where $f\_i\; :\; T\_i\; \backslash to\; S$ for every $i\; \backslash in\; I.$

The {{em|{{visible anchor|direct image bornology}}}} $\backslash mathcal\{A\}$ on $S$ determined by these maps is the weakest bornology on $S$ making each $f\_i\; :\; \backslash left(T\_i,\; \backslash mathcal\{B\}\_i\backslash right)\; \backslash to\; (S,\; \backslash mathcal\{A\})$ locally bounded.

If for each $i\; \backslash in\; I,$ $\backslash mathcal\{A\}\_i$ denotes the bornology generated by $f\backslash left(\backslash mathcal\{B\}\_i\backslash right),$ then this bornology is equal to the collection of all subsets $A$ of $S$ of the form $\backslash cup\_\{i\; \backslash in\; I\}\; A\_i$ where each $A\_i\; \backslash in\; \backslash mathcal\{A\}\_i$ and all but finitely many $A\_i$ are empty.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

Suppose that $(X,\; \backslash mathcal\{B\})$ is a bounded structure and $S$ be a subset of $X.$

The {{em|{{visible anchor|subspace bornology}}}} $\backslash mathcal\{A\}$ on $S$ is the finest bornology on $S$ making the inclusion map $(S,\; \backslash mathcal\{A\})\; \backslash to\; (X,\; \backslash mathcal\{B\})$ of $S$ into $X$ (defined by $s\; \backslash mapsto\; s$) locally bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

Let $\backslash left(X\_i,\; \backslash mathcal\{B\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ be an $I$-indexed family of bounded structures, let $X\; =\; \{\backslash textstyle\; \backslash prod\backslash limits\_\{i\; \backslash in\; I\}\; X\_i\},$ and for each $i\; \backslash in\; I,$ let $f\_i\; :\; X\; \backslash to\; X\_i$ denote the canonical projection.

The {{em|{{visible anchor|product bornology}}}} on $X$ is the inverse image bornology determined by the canonical projections $f\_i\; :\; X\; \backslash to\; X\_i.$

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 $\{\backslash textstyle\; \backslash left\backslash \{\backslash prod\backslash limits\_\{i\; \backslash in\; I\}\; B\_i\; ~:~\; B\_i\; \backslash in\; \backslash mathcal\{B\}\_i\; \backslash text\{\; for\; all\; \}\; i\; \backslash in\; I\backslash right\backslash \}\}.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

A subset of a topological space $X$ is called relatively compact if its closure is a compact subspace of $X.$

For any topological space $X$ in which singleton subsets are relatively compact (such as a T1 space), the set of all relatively compact subsets of $X$ form a bornology on $X$ called the {{em|{{visible anchor|compact bornology}}}} on $X.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

Every continuous map between T1 spaces is bounded with respect to their compact bornologies.

The set of relatively compact subsets of $\backslash R$ form a bornology on $\backslash R.$ A base for this bornology is given by all closed intervals of the form $[-n,\; n]$ for $n\; =\; 1,\; 2,\; 3,\; \backslash ldots.$

Given a metric space $(X,\; d),$ the {{em|metric bornology}} consists of all subsets $S\; \backslash subseteq\; X$ such that the supremum is finite.

Similarly, given a measure space $(X,\; \backslash Omega,\; \backslash mu),$ the family of all measurable subsets $S\; \backslash in\; \backslash Omega$ of finite measure (meaning ) form a bornology on $X.$

Suppose that $X$ is a topological space and $\backslash mathcal\{B\}$ is a bornology on $X.$

The bornology generated by the set of all topological interiors of sets in $\backslash mathcal\{B\}$ (that is, generated by $\backslash \{\backslash operatorname\{int\}\; B\; :\; B\; \backslash in\; \backslash mathcal\{B\}\backslash \}$ is called the {{em|{{visible anchor|interior bornology|text=interior}}}} of $\backslash mathcal\{B\}$ and is denoted by $\backslash operatorname\{int\}\; \backslash mathcal\{B\}.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

The bornology $\backslash mathcal\{B\}$ is called {{em|{{visible anchor|open bornology|text=open}}}} if $\backslash mathcal\{B\}\; =\; \backslash operatorname\{int\}\; \backslash mathcal\{B\}.$

The bornology generated by the set of all topological closures of sets in $\backslash mathcal\{B\}$ (that is, generated by $\backslash \{\backslash operatorname\{cl\}\; B\; :\; B\; \backslash in\; \backslash mathcal\{B\}\backslash \}$) is called the {{em|{{visible anchor|closure bornology|text=closure}}}} of $\backslash mathcal\{B\}$ and is denoted by $\backslash operatorname\{cl\}\; \backslash mathcal\{B\}.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

We necessarily have $\backslash operatorname\{int\}\; \backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{B\}\; \backslash subseteq\; \backslash operatorname\{cl\}\; \backslash mathcal\{B\}.$

The bornology $\backslash mathcal\{B\}$ is called {{em|{{visible anchor|closed bornology|text=closed}}}} if it satisfies any of the following equivalent conditions:

1. $\backslash mathcal\{B\}\; =\; \backslash operatorname\{cl\}\; \backslash mathcal\{B\};$
2. the closed subsets of $X$ generate $\backslash mathcal\{B\}$;{{sfn|Narici|Beckenstein|2011|pp=156-175}}
3. the closure of every $B\; \backslash in\; \backslash mathcal\{B\}$ belongs to $\backslash mathcal\{B\}.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

The bornology $\backslash mathcal\{B\}$ is called {{em|{{visible anchor|proper bornology|text=proper}}}} if $\backslash mathcal\{B\}$ is both open and closed.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

The topological space $X$ is called {{em|{{visible anchor|locally bounded topological space|text=locally $\backslash mathcal\{B\}$-bounded}}}} or just {{em|locally bounded}} if every $x\; \backslash in\; X$ has a neighborhood that belongs to $\backslash mathcal\{B\}.$

Every compact subset of a locally bounded topological space is bounded.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

{{See also|Vector bornology}}

If $X$ is a topological vector space (TVS) then the set of all bounded subsets of $X$ form a bornology (indeed, even a vector bornology) on $X$ called the {{em|{{visible anchor|von Neumann bornology|text=von Neumann bornology of $X$}}}}, the {{em|{{visible anchor|usual bornology}}}}, or simply {{em|{{visible anchor|the bornology}}}} of $X$ and is referred to as {{em|{{visible anchor|natural boundedness}}}}.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

In any locally convex TVS $X,$ the set of all closed bounded disks forms a base for the usual bornology of $X.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

A linear map between two bornological spaces is continuous if and only if it is bounded (with respect to the usual bornologies).

Suppose that $X$ is a commutative topological ring.

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

• {{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}}

{{reflist}}

• {{Hogbe-Nlend Bornologies and Functional Analysis}}
• {{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}}
• {{Kriegl Michor The Convenient Setting of Global Analysis}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Boundedness and bornology}}

{{Topological vector spaces}}

Category:Functional analysis

Category:Topological vector spaces