Bornological space

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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 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.[citation needed] The name was coined by Bourbaki[citation needed] after borné, the French word for "bounded".

Bornologies and bounded maps

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A bornology on a set   is a collection   of subsets of   that satisfy all the following conditions:

  1.   covers   that is,  ;
  2.   is stable under inclusions; that is, if   and   then  ;
  3.   is stable under finite unions; that is, if   then  ;

Elements of the collection   are called  -bounded or simply bounded sets if   is understood.[1] The pair   is called a bounded structure or a bornological set.[1]

A base or fundamental system 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 bornology generated by  [2]

If   and   are bornological sets then their product bornology on   is the bornology having as a base the collection of all sets of the form   where   and  [2] A subset of   is bounded in the product bornology if and only if its image under the canonical projections onto   and   are both bounded.

Bounded maps

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If   and   are bornological sets then a function   is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps  -bounded subsets of   to  -bounded subsets of   that is, if  [2] If in addition   is a bijection and   is also bounded then   is called a bornological isomorphism.

Vector bornologies

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Let   be a vector space over a field   where   has a bornology   A bornology   on   is called a vector bornology on   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:

  1.   is a vector bornology;
  2. Finite sums and balanced hulls of  -bounded sets are  -bounded;[2]
  3. 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).[2]

A vector bornology   is called a convex vector bornology 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 separated 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 convex vector bornology if   has a base consisting of convex sets.

Bornivorous subsets

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A subset   of   is called bornivorous and a bornivore 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.[3]

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[4]

Mackey convergence

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A sequence   in a TVS   is said to be Mackey convergent to   if there exists a sequence of positive real numbers   diverging to   such that   converges to   in  [5]

Bornology of a topological vector space

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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 the bornology or the von Neumann bornology 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).[4]

Induced topology

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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 topology induced by  .[4]

If   is a TVS then the bornological space associated with   is the vector space   endowed with the locally convex topology induced by the von Neumann bornology of  [4]

Theorem[4] — Let   and   be locally convex TVS and let   denote   endowed with the topology induced by von Neumann bornology of   Define   similarly. Then a linear map   is a bounded linear operator if and only if   is continuous.

Moreover, if   is bornological,   is Hausdorff, and   is continuous linear map then so is   If in addition   is also ultrabornological, then the continuity of   implies the continuity of   where   is the ultrabornological space associated with  

Quasi-bornological spaces

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Quasi-bornological spaces where introduced by S. Iyahen in 1968.[6]

A topological vector space (TVS)   with a continuous dual   is called a quasi-bornological space[6] if any of the following equivalent conditions holds:

  1. Every bounded linear operator from   into another TVS is continuous.[6]
  2. Every bounded linear operator from   into a complete metrizable TVS is continuous.[6][7]
  3. Every knot in a bornivorous string is a neighborhood of the origin.[6]

Every pseudometrizable TVS is quasi-bornological. [6] A TVS   in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.[8] If   is a quasi-bornological TVS then the finest locally convex topology on   that is coarser than   makes   into a locally convex bornological space.

Bornological space

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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 not quasi-bornological.[6]

A topological vector space (TVS)   with a continuous dual   is called a bornological space if it is locally convex and any of the following equivalent conditions holds:

  1. Every convex, balanced, and bornivorous set in   is a neighborhood of zero.[4]
  2. Every bounded linear operator from   into a locally convex TVS is continuous.[4]
    • Recall that a linear map is bounded if and only if it maps any sequence converging to   in the domain to a bounded subset of the codomain.[4] In particular, any linear map that is sequentially continuous at the origin is bounded.
  3. Every bounded linear operator from   into a seminormed space is continuous.[4]
  4. Every bounded linear operator from   into a Banach space is continuous.[4]

If   is a Hausdorff locally convex space then we may add to this list:[7]

  1. The locally convex topology induced by the von Neumann bornology on   is the same as    's given topology.
  2. Every bounded seminorm on   is continuous.[4]
  3. Any other Hausdorff locally convex topological vector space topology on   that has the same (von Neumann) bornology as   is necessarily coarser than  
  4.   is the inductive limit of normed spaces.[4]
  5.   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  ).[4]
  6.   carries the Mackey topology   and all bounded linear functionals on   are continuous.[4]
  7.   has both of the following properties:
    •   is convex-sequential or C-sequential, which means that every convex sequentially open subset of   is open,
    •   is sequentially bornological or S-bornological, which means that every convex and bornivorous subset of   is sequentially open.
    where a subset   of   is called sequentially open if every sequence converging to   eventually belongs to  

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,[4] 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:

  • 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.

Sufficient conditions

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Mackey–Ulam theorem[9] — The product of a collection   locally convex bornological spaces is bornological if and only if   does not admit an Ulam measure.

As a consequent of the Mackey–Ulam theorem, "for all practical purposes, the product of bornological spaces is bornological."[9]

The following topological vector spaces are all bornological:

  • Any locally convex pseudometrizable TVS is bornological.[4][10]
  • Any strict inductive limit of bornological spaces, in particular any strict LF-space, is bornological.
    • This shows that there are bornological spaces that are not metrizable.
  • A countable product of locally convex bornological spaces is bornological.[11][10]
  • Quotients of Hausdorff locally convex bornological spaces are bornological.[10]
  • The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.[10]
  • Fréchet Montel spaces have bornological strong duals.
  • The strong dual of every reflexive Fréchet space is bornological.[12]
  • If the strong dual of a metrizable locally convex space is separable, then it is bornological.[12]
  • A vector subspace of a Hausdorff locally convex bornological space   that has finite codimension in   is bornological.[4][10]
  • The finest locally convex topology on a vector space is bornological.[4]
Counterexamples

There exists a bornological LB-space whose strong bidual is not bornological.[13]

A closed vector subspace of a locally convex bornological space is not necessarily bornological.[4][14] There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.[4]

Bornological spaces need not be barrelled and barrelled spaces need not be bornological.[4] Because every locally convex ultrabornological space is barrelled,[4] it follows that a bornological space is not necessarily ultrabornological.

Properties

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  • The strong dual space of a locally convex bornological space is complete.[4]
  • Every locally convex bornological space is infrabarrelled.[4]
  • Every Hausdorff sequentially complete bornological TVS is ultrabornological.[4]
    • Thus every complete Hausdorff bornological space is ultrabornological.
    • In particular, every Fréchet space is ultrabornological.[4]
  • The finite product of locally convex ultrabornological spaces is ultrabornological.[4]
  • Every Hausdorff bornological space is quasi-barrelled.[15]
  • Given a bornological space   with continuous dual   the topology of   coincides with the Mackey topology  
  • 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.
  • Every bornological space is the inductive limit of normed spaces (and Banach spaces if the space is also quasi-complete).
  • Let   be a metrizable locally convex space with continuous dual   Then the following are equivalent:
    1.   is bornological.
    2.   is quasi-barrelled.
    3.   is barrelled.
    4.   is a distinguished space.
  • If   is a linear map between locally convex spaces and if   is bornological, then the following are equivalent:
    1.   is continuous.
    2.   is sequentially continuous.[4]
    3. For every set   that's bounded in     is bounded.
    4. If   is a null sequence in   then   is a null sequence in  
    5. If   is a Mackey convergent null sequence in   then   is a bounded subset of  
  • 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.[4]
    • In particular, the strong dual of a locally convex bornological space is complete.[4] However, it need not be bornological.
Subsets
  • In a locally convex bornological space, every convex bornivorous set   is a neighborhood of   (  is not required to be a disk).[4]
  • Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[4]
  • Closed vector subspaces of bornological space need not be bornological.[4]

Ultrabornological spaces

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A disk in a topological vector space   is called infrabornivorous 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 ultrabornological if any of the following equivalent conditions hold:

  1. Every infrabornivorous disk is a neighborhood of the origin.
  2.   is the inductive limit of the spaces   as   varies over all compact disks in  
  3. A seminorm on   that is bounded on each Banach disk is necessarily continuous.
  4. For every locally convex space   and every linear map   if   is bounded on each Banach disk then   is continuous.
  5. For every Banach space   and every linear map   if   is bounded on each Banach disk then   is continuous.

Properties

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

See also

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References

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  1. ^ a b Narici & Beckenstein 2011, p. 168.
  2. ^ a b c d e Narici & Beckenstein 2011, pp. 156–175.
  3. ^ Wilansky 2013, p. 50.
  4. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag Narici & Beckenstein 2011, pp. 441–457.
  5. ^ Swartz 1992, pp. 15–16.
  6. ^ a b c d e f g Narici & Beckenstein 2011, pp. 453–454.
  7. ^ a b Adasch, Ernst & Keim 1978, pp. 60–61.
  8. ^ Wilansky 2013, p. 48.
  9. ^ a b Narici & Beckenstein 2011, p. 450.
  10. ^ a b c d e Adasch, Ernst & Keim 1978, pp. 60–65.
  11. ^ Narici & Beckenstein 2011, p. 453.
  12. ^ a b Schaefer & Wolff 1999, p. 144.
  13. ^ Khaleelulla 1982, pp. 28–63.
  14. ^ Schaefer & Wolff 1999, pp. 103–110.
  15. ^ Adasch, Ernst & Keim 1978, pp. 70–73.

Bibliography

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  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
  • Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
  • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
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