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| 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]].
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| ==Bornological sets==
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| Let ''X'' be any set. A '''bornology''' on ''X'' is a collection ''B'' of subsets of ''X'' such that
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| * ''B'' covers ''X'', i.e. <math>X = \bigcup B;</math>
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| * ''B'' is stable under inclusions, i.e. if ''A'' ∈ ''B'' and ''A′'' ⊆ ''A'', then ''A′'' ∈ ''B'';
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| * ''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>
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| Elements of the collection ''B'' are called '''bounded sets''', and the pair (''X'', ''B'') is called a '''bornological set'''.
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| 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>.
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| ===Examples===
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| * For any set ''X'', the [[discrete topology]] of ''X'' is a bornology.
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| * For any set ''X'', the set of finite (or countably infinite) subsets of ''X'' is a bornology.
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| * For any topological space ''X'' that is ''T1'', the set of subsets of ''X'' with [[compact space|compact]] [[closure (topology)|closure]] is a bornology.
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| ==Bounded maps==
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| 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'''.
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| Examples:
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| * 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.
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| Theorems:
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| * 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:
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| ** ''u'' is a bounded map,
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| ** ''u''takes bounded disks to bounded disks,
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| ** For every bornivorous disk ''D'' in ''Y'', <math>u^{-1}(D)</math> is bornivorous.
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| ==Vector bornologies==
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| 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.
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| ===Bornology of a topological vector space===
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| 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>.
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| The set of all bounded subsets of <math>X</math> is called the '''bornology''' or the '''Von-Neumann bornology''' of <math>X</math>.
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| ===Induced topology===
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| 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>.
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| ==Bornological spaces==
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| 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:
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| * The locally convex topology induced by the von-Neumann bornology on <math>X</math> is the same as <math>X</math>'s [[initial topology]],
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| * Every bounded [[semi-norm]] on <math>X</math> is continuous,
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| * For all locally convex spaces ''Y'', every [[bounded linear operator]]s from <math>X</math> into <math>Y</math> is [[continuous linear operator|continuous]].
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| * ''X'' is the inductive limit of normed spaces.
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| * ''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'').
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| * Every convex, balanced, and bornivorous set in <math>X</math> is a neighborhood of <math>0</math>.
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| * ''X'' caries the Mackey topology <math>\tau(X, X')</math> and all bounded linear functionals on ''X'' are continuous.
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| * <math>X</math> has both of the following properties:
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| ** <math>X</math> is '''convex-sequential''' or '''C-sequential''', which means that every convex sequentially open subset of <math>X</math> is open,
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| ** <math>X</math> is '''sequentially-bornological''' or '''S-bornological''', which means that every convex and bornivorous subset of <math>X</math> is sequentially open.
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| where a subset ''A'' of <math>X</math> is called '''sequentially open''' if every sequence converging to ''0'' eventually belongs to ''A''.
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| ===Examples===
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| The following topological vector spaces are all bornological:
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| * Any [[metrisable]] locally convex space is bornological. In particular, any [[Fréchet space]].
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| * Any ''LF''-space (i.e. any locally convex space that is the strict inductive limit of [[Fréchet space]]s).
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| * Separated quotients of bornological spaces are bornological.
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| * The locally convex direct sum and inductive limit of bornological spaces is bornological.
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| * [[Frechet space|Frechet]] [[Montel space|Montel]] have a bornological strong dual.
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| ===Properties===
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| * Given a bornological space ''X'' with [[continuous dual]] ''X′'', then the topology of ''X'' coincides with the [[Mackey topology]] τ(''X'',''X′'').
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| ** In particular, bornological spaces are [[Mackey space]]s.
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| * 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.
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| * Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete).
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| * Let <math>X</math> be a metrizable locally convex space with continuous dual <math>X'</math>. Then the following are equivalent:
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| ** <math>\beta(X', X)</math> is bornological,
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| ** <math>\beta(X', X)</math> is [[barrelled space|quasi-barrelled]],
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| ** <math>\beta(X', X)</math> is [[barrelled space|barrelled]],
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| ** <math>X</math> is a [[distingushed space]].
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| * 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:
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| ** <math>u</math> is continuous,
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| ** for every set <math> B \sub X</math> that's bounded in <math>X</math>, <math>u(B)</math> is bounded,
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| ** 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>.
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| * The strong dual of a bornological space is complete, but it need not be bornological.
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| * Closed subspaces of bornological space need not be bornological.
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| ==Banach Disks==
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| 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.
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| 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'''.
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| Suppose that ''X'' is a locally convex Hausdorff space and that ''D'' is a bounded disk in ''X''. Then
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| * 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>.
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| ===Examples=== | |
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| * Any closed and bounded disk in a Banach space is a Banach disk.
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| * 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.
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| ==Ultrabornological spaces==
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| 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:
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| * every infrabornivorous disk is a neighborhood of 0,
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| * ''X'' be the inductive limit of the spaces <math>X_D</math> as ''D'' varies over all compact disks in ''X'',
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| * A seminorm on ''X'' that is bounded on each Banach disk is necessarily continuous,
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| * 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.
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| * 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.
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| ===Properties===
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| * The finite product of ultrabornological spaces is ultrabornological.
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| * Inductive limits of ultrabornological spaces are ultrabornological.
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| == See also ==
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| * [[Space of linear maps]]
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| == References ==
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| {{reflist}}
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| * {{cite book
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| | last = Hogbe-Nlend
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| | first = Henri
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| | title = Bornologies and functional analysis
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| | publisher = North-Holland Publishing Co.
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| | location = Amsterdam
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| | year = 1977
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| | pages = xii+144
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| | isbn = 0-7204-0712-5
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| | mr = 0500064
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| }}
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| * {{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 }}
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| * {{Cite isbn|9783540115656|pages = 29-33, 49, 104}}
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| {{Functional Analysis}}
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| [[Category:Topological vector spaces]]
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| {{mathanalysis-stub}}
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