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 sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.
Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after , the French word for "bounded".
A on a set is a collection of subsets of that satisfy all the following conditions: <ol> <li> covers that is, ;</li> <li> is stable under inclusions; that is, if and then ;</li> <li> is stable under finite unions; that is, if then ;</li> </ol> Elements of the collection are called or simply if is understood. The pair is called a or a .
A or of a bornology is a subset of such that each element of is a subset of some element of Given a collection of subsets of the smallest bornology containing is called the
If and are bornological sets then their on is the bornology having as a base the collection of all sets of the form where and A subset of is bounded in the product bornology if and only if its image under the canonical projections onto and are both bounded.
If and are bornological sets then a function is said to be a or a (with respect to these bornologies) if it maps -bounded subsets of to -bounded subsets of that is, if If in addition is a bijection and is also bounded then is called a .
Let be a vector space over a field where has a bornology A bornology on is called a if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).
If is a topological vector space (TVS) and is a bornology on then the following are equivalent: <ol> <li> is a vector bornology;</li> <li>Finite sums and balanced hulls of -bounded sets are -bounded;</li> <li>The scalar multiplication map defined by and the addition map defined by are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).</li> </ol>
A vector bornology is called a if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then And a vector bornology is called if the only bounded vector subspace of is the 0-dimensional trivial space
Usually, is either the real or complex numbers, in which case a vector bornology on will be called a if has a base consisting of convex sets.
A subset of is called and a if it absorbs every bounded set.
In a vector bornology, is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology is bornivorous if it absorbs every bounded disk.
Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.
Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.
A sequence in a TVS is said to be if there exists a sequence of positive real numbers diverging to such that converges to in
Every topological vector space at least on a non discrete valued field gives a bornology on by defining a subset to be bounded (or von-Neumann bounded), if and only if for all open sets containing zero there exists a with If is a locally convex topological vector space then is bounded if and only if all continuous semi-norms on are bounded on
The set of all bounded subsets of a topological vector space is called or of
If is a locally convex topological vector space, then an absorbing disk in is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).
If is a convex vector bornology on a vector space then the collection of all convex balanced subsets of that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on called the .
If is a TVS then the is the vector space endowed with the locally convex topology induced by the von Neumann bornology of
Quasi-bornological spaces where introduced by S. Iyahen in 1968.
A topological vector space (TVS) with a continuous dual is called a if any of the following equivalent conditions holds:
<ol> <li>Every bounded linear operator from into another TVS is continuous.</li> <li>Every bounded linear operator from into a complete metrizable TVS is continuous.</li> <li>Every knot in a bornivorous string is a neighborhood of the origin.</li> </ol>
Every pseudometrizable TVS is quasi-bornological. A TVS in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space. If is a quasi-bornological TVS then the finest locally convex topology on that is coarser than makes into a locally convex bornological space.
In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.
Every locally convex quasi-bornological space is bornological but there exist bornological spaces that are quasi-bornological.
A topological vector space (TVS) with a continuous dual is called a if it is locally convex and any of the following equivalent conditions holds:
<ol> <li>Every convex, balanced, and bornivorous set in is a neighborhood of zero.</li> <li>Every bounded linear operator from into a locally convex TVS is continuous.
<li>Every bounded linear operator from into a seminormed space is continuous.</li> <li>Every bounded linear operator from into a Banach space is continuous.</li> </ol>
If is a Hausdorff locally convex space then we may add to this list: <ol start=5> <li>The locally convex topology induced by the von Neumann bornology on is the same as 's given topology.</li> <li>Every bounded seminorm on is continuous.</li> <li>Any other Hausdorff locally convex topological vector space topology on that has the same (von Neumann) bornology as is necessarily coarser than </li> <li> is the inductive limit of normed spaces.</li> <li> is the inductive limit of the normed spaces as varies over the closed and bounded disks of (or as varies over the bounded disks of ).</li> <li> carries the Mackey topology and all bounded linear functionals on are continuous.</li> <li> has both of the following properties:
where a subset of is called if every sequence converging to eventually belongs to </li> </ol>
Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous, where recall that a linear operator is sequentially continuous if and only if it is sequentially continuous at the origin. Thus for linear maps from a bornological space into a locally convex space, continuity is equivalent to sequential continuity at the origin. More generally, we even have the following:
<ul> <li>Any linear map from a locally convex bornological space into a locally convex space that maps null sequences in to bounded subsets of is necessarily continuous.</li> </ul>
As a consequent of the MackeyâÂÂUlam theorem, "for all practical purposes, the product of bornological spaces is bornological."
The following topological vector spaces are all bornological: <ul> <li>Any locally convex pseudometrizable TVS is bornological.
<li>Any strict inductive limit of bornological spaces, in particular any strict LF-space, is bornological.
<li>A countable product of locally convex bornological spaces is bornological.</li> <li>Quotients of Hausdorff locally convex bornological spaces are bornological.</li> <li>The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.</li> <li>Fréchet Montel spaces have bornological strong duals.</li> <li>The strong dual of every reflexive Fréchet space is bornological.</li> <li>If the strong dual of a metrizable locally convex space is separable, then it is bornological.</li> <li>A vector subspace of a Hausdorff locally convex bornological space that has finite codimension in is bornological.</li> <li>The finest locally convex topology on a vector space is bornological.</li> </ul>
There exists a bornological LB-space whose strong bidual is bornological.
A closed vector subspace of a locally convex bornological space is not necessarily bornological. There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.
Bornological spaces need not be barrelled and barrelled spaces need not be bornological. Because every locally convex ultrabornological space is barrelled, it follows that a bornological space is not necessarily ultrabornological.
<ul> <li>The strong dual space of a locally convex bornological space is complete.</li> <li>Every locally convex bornological space is infrabarrelled.</li> <li>Every Hausdorff sequentially complete bornological TVS is ultrabornological.
<li>The finite product of locally convex ultrabornological spaces is ultrabornological.</li> <li>Every Hausdorff bornological space is quasi-barrelled.</li> <li>Given a bornological space with continuous dual the topology of coincides with the Mackey topology
<li>Every quasi-complete (i.e. all closed and bounded subsets are complete) bornological space is barrelled. There exist, however, bornological spaces that are not barrelled.</li> <li>Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete).</li> <li> Let be a metrizable locally convex space with continuous dual Then the following are equivalent: <ol> <li> is bornological.</li> <li> is quasi-barrelled.</li> <li> is barrelled.</li> <li> is a distinguished space.</li> </ol> </li> <li>If is a linear map between locally convex spaces and if is bornological, then the following are equivalent: <ol> <li> is continuous.</li> <li> is sequentially continuous.</li> <li>For every set that's bounded in is bounded.</li> <li>If is a null sequence in then is a null sequence in </li> <li>If is a Mackey convergent null sequence in then is a bounded subset of </li> </ol> </li> <li>Suppose that and are locally convex TVSs and that the space of continuous linear maps is endowed with the topology of uniform convergence on bounded subsets of If is a bornological space and if is complete then is a complete TVS.
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<ul> <li>In a locally convex bornological space, every convex bornivorous set is a neighborhood of ( is required to be a disk).</li> <li>Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.</li> <li>Closed vector subspaces of bornological space need not be bornological.</li> </ul>
A disk in a topological vector space is called if it absorbs all Banach disks.
If 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 if any of the following equivalent conditions hold: <ol> <li>Every infrabornivorous disk is a neighborhood of the origin.</li> <li> is the inductive limit of the spaces as varies over all compact disks in </li> <li>A seminorm on that is bounded on each Banach disk is necessarily continuous.</li> <li>For every locally convex space and every linear map if is bounded on each Banach disk then is continuous.</li> <li>For every Banach space and every linear map if is bounded on each Banach disk then is continuous.</li> </ol>
The finite product of ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological.