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5.64.169.180: Undid revision 587350202 by 49.204.138.154 (talk) which mangled the section

In [[mathematics]], particularly in [[functional analysis]], a '''bornological space''' is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of [[bounded set|sets]] and [[bounded function|functions]], in the same way that a [[topological space]] possesses the minimum amount of structure needed to address questions of [[continuous function|continuity]]. Bornological spaces were first studied by Mackey and their name was given by [[Bourbaki]].

==Bornological sets==

Let ''X'' be any set. A '''bornology''' on ''X'' is a collection ''B'' of subsets of ''X'' such that
* ''B'' covers ''X'', i.e. <math>X = \bigcup B;</math>
* ''B'' is stable under inclusions, i.e. if ''A'' ∈ ''B'' and ''A′'' ⊆ ''A'', then ''A′'' ∈ ''B'';
* ''B'' is stable under finite unions, i.e. if ''B''<sub>1</sub>, ..., ''B''<sub>''n''</sub> ∈ ''B'', then <math>\bigcup_{i = 1}^{n} B_{i} \in B.</math>
Elements of the collection ''B'' are called '''bounded sets''', and the pair (''X'', ''B'') is called a '''bornological set'''.

A '''base of the bornology''' ''B'' is a subset <math>B_0</math> of ''B'' such that each element of ''B'' is a subset of an element of <math>B_0</math>.

===Examples===

* For any set ''X'', the [[discrete topology]] of ''X'' is a bornology.
* For any set ''X'', the set of finite (or countably infinite) subsets of ''X'' is a bornology.
* For any topological space ''X'' that is ''T1'', the set of subsets of ''X'' with [[compact space|compact]] [[closure (topology)|closure]] is a bornology.

==Bounded maps==

If <math>B_1</math> and <math>B_2</math> are two bornologies over the spaces <math>X</math> and <math>Y</math>, respectively, and if <math>f\colon X \rightarrow Y</math> is a function, then we say that <math>f</math> is a '''bounded map''' if it maps <math>B_1</math>-bounded sets in <math>X</math> to <math>B_2</math>-bounded sets in <math>Y</math>. If in addition <math>f</math> is a bijection and <math>f^{-1}</math> is also bounded then we say that <math>f</math> is a '''bornological isomorphism'''.

Examples:
* If <math>X</math> and <math>Y</math> are any two topological vector spaces (they need not even be Hausdorff) and if <math>f\colon X \rightarrow Y</math> is a continuous linear operator between them, then <math>f</math> is a bounded linear operator (when <math>X</math> and <math>Y</math> have their von-Neumann bornologies). The converse is in general false.

Theorems:
* Suppose that ''X'' and ''Y'' are locally convex spaces and that <math>u : X \to Y</math> is a linear map. Then the following are equivalent:
** ''u'' is a bounded map,
** ''u''takes bounded disks to bounded disks,
** For every bornivorous disk ''D'' in ''Y'', <math>u^{-1}(D)</math> is bornivorous.

==Vector bornologies==

If <math>X</math> is a vector space over a field ''K'' and then a '''vector bornology on <math>X</math>''' is a bornology ''B'' on <math>X</math> that 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 in addition ''B'' is stable under the formation of [[convex hull]]s (i.e. the convex hull of a bounded set is bounded) then ''B'' is called a '''convex vector bornology'''. And if the only bounded subspace of <math>X</math> is the trivial subspace (i.e. the space consisting only of <math>0</math>) then it is called '''separated'''. A subset ''A'' of ''B'' is called '''bornivorous''' if it [[absorbing set|absorbs]] every bounded set. In a vector bornology, ''A'' is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology ''A'' is bornivorous if it absorbs every bounded disk.

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

Every [[topological vector space]] <math>X</math> gives a bornology on X by defining a subset <math>B\subseteq X</math> to be [[Bounded set (topological vector space)|bounded]] (or von-Neumann bounded), if and only if for all open sets <math>U\subseteq X</math>containing zero there exists a <math>\lambda>0</math> with <math>B\subseteq\lambda U</math>. If <math>X</math> is a [[locally convex topological vector space]] then <math>B\subseteq X</math> is bounded if and only if all continuous semi-norms on <math>X</math> are bounded on <math>A</math>.

The set of all bounded subsets of <math>X</math> is called the '''bornology''' or the '''Von-Neumann bornology''' of <math>X</math>.

===Induced topology===

Suppose that we start with a vector space <math>X</math> and convex vector bornology ''B'' on <math>X</math>. If we let ''T'' denote the collection of all sets that are convex, balanced, and bornivorous then ''T'' forms neighborhood basis at 0 for a locally convex topology on <math>X</math> that is compatible with the vector space structure of <Math>X</math>.

==Bornological spaces==

In functional analysis, a bornological space is a [[locally convex topological vector space]] whose topology can be recovered from its bornology in a natural way. Explicitly, a Hausdorff [[locally convex space]] <math>X</math> with [[continuous dual]] <math>X'</math> is called a bornological space if any one of the following equivalent conditions holds:
* The locally convex topology induced by the von-Neumann bornology on <math>X</math> is the same as <math>X</math>'s [[initial topology]],
* Every bounded [[semi-norm]] on <math>X</math> is continuous,
* For all locally convex spaces ''Y'', every [[bounded linear operator]]s from <math>X</math> into <math>Y</math> is [[continuous linear operator|continuous]].
* ''X'' is the inductive limit of normed spaces.
* ''X'' is the inductive limit of the normed spaces ''X_D'' as ''D'' varies over the closed and bounded disks of ''X'' (or as ''D'' varies over the bounded disks of ''X'').
* Every convex, balanced, and bornivorous set in <math>X</math> is a neighborhood of <math>0</math>.
* ''X'' caries the Mackey topology <math>\tau(X, X')</math> and all bounded linear functionals on ''X'' are continuous.
* <math>X</math> has both of the following properties:
** <math>X</math> is '''convex-sequential''' or '''C-sequential''', which means that every convex sequentially open subset of <math>X</math> is open,
** <math>X</math> is '''sequentially-bornological''' or '''S-bornological''', which means that every convex and bornivorous subset of <math>X</math> is sequentially open.
where a subset ''A'' of <math>X</math> is called '''sequentially open''' if every sequence converging to ''0'' eventually belongs to ''A''.

===Examples===
The following topological vector spaces are all bornological:
* Any [[metrisable]] locally convex space is bornological. In particular, any [[Fréchet space]].
* Any ''LF''-space (i.e. any locally convex space that is the strict inductive limit of [[Fréchet space]]s).
* Separated quotients of bornological spaces are bornological.
* The locally convex direct sum and inductive limit of bornological spaces is bornological.
* [[Frechet space|Frechet]] [[Montel space|Montel]] have a bornological strong dual.

===Properties===
* Given a bornological space ''X'' with [[continuous dual]] ''X′'', then the topology of ''X'' coincides with the [[Mackey topology]] τ(''X'',''X′'').
** In particular, bornological spaces are [[Mackey space]]s.
* Every [[quasi-complete]] (i.e. all closed and bounded subsets are complete) bornological space is [[barrelled space|barrelled]]. There exist, however, bornological spaces that are not barrelled.
* Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete).
* Let <math>X</math> be a metrizable locally convex space with continuous dual <math>X'</math>. Then the following are equivalent:
** <math>\beta(X', X)</math> is bornological,
** <math>\beta(X', X)</math> is [[barrelled space|quasi-barrelled]],
** <math>\beta(X', X)</math> is [[barrelled space|barrelled]],
** <math>X</math> is a [[distingushed space]].
* If <math>X</math> is bornological, <math>Y</math> is a locally convex TVS, and <math>u : X \to Y</math> is a linear map, then the following are equivalent:
** <math>u</math> is continuous,
** for every set <math> B \sub X</math> that's bounded in <math>X</math>, <math>u(B)</math> is bounded,
** If <math>(x_n) \sub X</math> is a null sequence in <math>X</math> then <math>(u(x_n))</math> is a null sequence in <math>Y</math>.
* The strong dual of a bornological space is complete, but it need not be bornological.
* Closed subspaces of bornological space need not be bornological.

==Banach Disks==

Suppose that ''X'' is a topological vector space. Then we say that a subset ''D'' of ''X'' is a disk if it is convex and balanced. The disk ''D'' is absorbing in the space ''span(D)'' and so its [[Minkowski functional]] forms a seminorm on this space, which is denoted by <math>\mu_D</math> or by <math>p_D</math>. When we give ''span(D)'' the topology induced by this seminorm we denote the resulting topological vector space by <math>X_D</math>. A basis of neighborhoods of ''0'' of this space consists of all sets of the form ''r D'' where ''r'' ranges over all positive real numbers.

This space is not necessarily Hausdorff as is the case, for instance, if we let <math>X = \mathbb{R}^2</math> and ''D'' be the ''x''-axis. However, if ''D'' is a bounded disk and if ''X'' is Hausdorff then we have that <math>\mu_D</math> is a norm and so that <math>X_D</math> is a normed space. If ''D'' is a bounded sequentially complete disk and''X'' is Hausdorff then the space <math>X_D</math> is in fact a Banach space. And bounded disk in ''X'' for which <math>X_D</math> is a Banach space is called a '''Banach disk''', '''infracomplete''', or a '''bounded completant'''.

Suppose that ''X'' is a locally convex Hausdorff space and that ''D'' is a bounded disk in ''X''. Then
* If ''D'' is complete in ''X'' and ''T'' is a Barrell in ''X'' then there is a number ''r > 0'' such that <math>B \subseteq r T</math>.

===Examples===

* Any closed and bounded disk in a Banach space is a Banach disk.

* If ''U'' is a convex balanced closed neighborhood of ''0'' in ''X'' then we can place on ''X'' the topological vector space topology induced by the neighborhoods ''r U'' where ''r > 0'' ranges over the positive real numbers. When ''X'' has this topology it is denoted by ''X_U''. However, this topology is not necessarily Hausdorff or complete so we denote the completion of the Hausdorff space <math>X_U/\ker(\mu_U)</math> by <math>\hat{X}_U</math> so that <math>\hat{X}_U</math> is a complete Hausdorff space and <math>\mu_U</math> is a norm on this space so that <math>\hat{X}_U</math> is a Banach space. If we let <math>D'</math> be the polar of ''U'' then <math>D'</math> is a weakly compact bounded equicontinuous disk in <math>X^*</math> and so is infracomplete.

==Ultrabornological spaces==

A disk in a topological vector space ''X'' is called '''infrabornivorous''' if it absorbs all Banach disks. If ''X'' is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks. A locally convex space is called '''ultrabornological''' if any of the following conditions hold:
* every infrabornivorous disk is a neighborhood of 0,
* ''X'' be the inductive limit of the spaces <math>X_D</math> as ''D'' varies over all compact disks in ''X'',
* A seminorm on ''X'' that is bounded on each Banach disk is necessarily continuous,
* For every locally convex space ''Y'' and every linear map <math>u : X \to Y</math>, if ''u'' is bounded on each Banach disk then ''u'' is continuous.
* For every Banach space ''Y'' and every linear map <math>u : X \to Y</math>, if ''u'' is bounded on each Banach disk then ''u'' is continuous.

===Properties===

* The finite product of ultrabornological spaces is ultrabornological.
* Inductive limits of ultrabornological spaces are ultrabornological.

== See also ==
* [[Space of linear maps]]

== References ==

{{reflist}}

* {{cite book
| last = Hogbe-Nlend
| first = Henri
| title = Bornologies and functional analysis
| publisher = North-Holland Publishing Co.
| location = Amsterdam
| year = 1977
| pages = xii+144
| isbn = 0-7204-0712-5
| mr = 0500064
}}
* {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | year=1970 | isbn=0-387-05380-8 | pages=61–63 }}
* {{Cite isbn|9783540115656|pages = 29-33, 49, 104}}

{{Functional Analysis}}

[[Category:Topological vector spaces]]

{{mathanalysis-stub}}

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