Bornological space

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 that 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 Mackey. The name was coined by Bourbaki after borné, the French word for "bounded".

Bornologies and bounded maps
Main article: Bornology

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

  1. ℬ covers X, i.e. X = ∪ ℬ;
  2. ℬ is stable under inclusions, i.e. if B ∈ ℬ and A′ ⊆ B, then A′ ∈ ℬ;
  3. ℬ is stable under finite unions, i.e. if B1, ..., Bn ∈ ℬ, then B1 ∪ ⋅⋅⋅ ∪ Bn ∈ ℬ

Elements of the collection ℬ are usually called ℬ-bounded or simply bounded sets. The pair (X, ℬ) is called a bounded structure or a bornological set.

A base of the bornology ℬ is a subset ℬ0 of ℬ such that each element of ℬ is a subset of an element of ℬ0.

Bounded maps

If B1 and B2 are two bornologies over the spaces X and Y, respectively, and if f : X → Y is a function, then we say that f is a locally bounded map or a bounded map if it maps B1-bounded sets in X to B2-bounded sets in Y. If in addition f is a bijection and f −1 is also bounded then we say that f is a bornological isomorphism.

Vector bornologies
Main article: Vector bornology

If X is a vector space over a field 𝕂 then a vector bornology on X is a bornology ℬ on X that 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 X is a topological vector space (TVS) and ℬ is a bornology on X, then the following are equivalent:

  1. ℬ is a vector bornology;
  2. Finite sums and balanced hulls of ℬ-bounded sets are ℬ-bounded;[1]
  3. The scalar multiplication map 𝕂 ×; X → X defined by (s, x) ↦ sx and the addition map X ×; X → X defined by (x, y) ↦ x + y, are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).[1]

A vector bornology ℬ 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 ℬ is called a . And a vector bornology ℬ is called separated if the only bounded vector subspace of X is the 0-dimensional trivial space { 0  }.

Bornivorous subsets

A subset A of X is called bornivorous and a bornivore if it 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.

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.[2]

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[3]

Mackey convergence

A sequence x• = (xi)∞
i=1 in a TVS X is said to be Mackey convergent to 0 if there exists a sequence of positive real numbers r• = (ri)∞
i=1 diverging to ∞ such that (rixi)∞
i=1 converges to 0 in X.[4]

Bornology of a topological vector space

Every topological vector space X, at least on a non discrete valued field gives a bornology on X by defining a subset B ⊆ X to be bounded (or von-Neumann bounded), if and only if for all open sets U ⊆ X containing zero there exists a r > 0 with B ⊆ rU. If X is a locally convex topological vector space then B ⊆ X is bounded if and only if all continuous semi-norms on X are bounded on B.

The set of all bounded subsets of a topological vector space X is called the bornology or the von Neumann bornology of X.

If X is a locally convex topological vector space, then an absorbing disk D in X is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).[3]

Induced topology

If ℬ is a convex vector bornology on a vector space X, then the collection 𝒩ℬ(0) of all convex balanced subsets of X that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on X called the topology induced by ℬ.[3]

If (X, τ) is a TVS then the bornological space associated with X is the vector space X endowed with the locally convex topology induced by the von Neumann bornology of (X, τ).[3]

Theorem[3] — Let X and Y be locally convex TVS and let Xb denote X endowed with the topology induced by von Neumann bornology of X. Define Yb similarly. Then a linear map L : X → Y is a bounded linear operator if and only if L : Xb → Y is continuous.

Moreover, if X is bornological, Y is Hausdorff, and L : X → Y is continuous linear map then so is L : X → Yb. If in addition X is also ultrabornological, then the continuity of L : X → Y implies the continuity of L : X → Yub, where Yub is the ultrabornological space associated with Y.

Bornological spaces

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.

Quasi-bornological spaces

Quasi-bornological spaces where introduced by S. Iyahen in 1968.[5]

A topological vector space (TVS) (X, τ) with a continuous dual X ' is called a quasi-bornological space[5] if any of the following equivalent conditions holds:

  1. Every bounded linear operator from X into another TVS is continuous.[5]
  2. Every bounded linear operator from X into a complete metrizable TVS is continuous.[5][6]
  3. Every knot in a bornivorous string is a neighborhood of the origin.[5]

Every pseudometrizable TVS is quasi-bornological. [5] A TVS (X, τ) in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.[7] If X is a quasi-bornological TVS then the finest locally convex topology on X that is coarser than τ makes X into a locally convex bornological space.

Bornological space

Note that every locally convex quasi-bornological space is bornological but there exist bornological spaces that are not quasi-bornological.[5]

A topological vector space (TVS) (X, τ) with a continuous dual X ' 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 X is a neighborhood of zero.[3]
  2. Every bounded linear operator from X into a locally convex TVS is continuous.[3]
    • Recall that a linear map is bounded if and only if it maps any sequence converging to 0 in the domain to a bounded subset of the codomain.[3] In particular, any linear map that is sequentially continuous at the origin is bounded.
  3. Every bounded linear operator from X into a seminormed space is continuous.[3]
  4. Every bounded linear operator from X into a Banach space is continuous.[3]

If X is a Hausdorff locally convex space then we may add to this list:[6]

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

    where a subset A of X is called sequentially open if every sequence converging to 0 eventually belongs to A.

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,[3] 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 F : X → Y from a locally convex bornological space into a locally convex space Y that maps null sequences in X to bounded subsets of Y is necessarily continuous.

Sufficient conditions

Mackey-Ulam theorem[8] — The product of a collection X• = (Xi)i ∈ I locally convex bornological spaces is bornological if and only if I 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."[8]

The following topological vector spaces are all bornological:

Counter examples
  • There exists a bornological LB-space whose strong bidual is not bornological.[12]
  • A closed vector subspace of a locally convex bornological space is not necessarily bornological.[3][13]
  • Bornological spaces need not be barrelled and barrelled spaces need not be bornological.[3]
    • Since every locally convex ultrabornological space is barrelled,[3] it follows that a bornological space is not necessarily ultrabornological.
  • There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.[3]

Properties
  • The strong dual space of a locally convex bornological space is complete.[3]
  • Every locally convex bornological space is infrabarrelled.[3]
  • Every Hausdorff sequentially complete bornological TVS is ultrabornological.[3]
    • Thus every compete Hausdorff bornological space is ultrabornological.
    • In particular, every Fréchet space is ultrabornological.[3]
  • The finite product of locally convex ultrabornological spaces is ultrabornological.[3]
  • Every Hausdorff bornological space is quasi-barrelled.[14]
  • Given a bornological space X with continuous dual X′, the topology of X coincides with the Mackey topology τ(X,X′).
  • 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 X 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. X is a distinguished space.
  • If L : X → Y is a linear map between locally convex spaces and if X is bornological, then the following are equivalent:
    1. L : X → Y is continuous.
    2. L : X → Y is sequentially continuous.[3]
    3. For every set B ⊆ X that's bounded in X, L(B) is bounded.
    4. If (xn) ⊆ X is a null sequence in X then (L(xn)) is a null sequence in Y.
    5. If (xn) ⊆ X is a Mackey convergent null sequence in X then (L(xn)) is a bounded subset of Y.
  • Suppose that X and Y are locally convex TVSs and that the space of continuous linear maps Lb(X; Y) is endowed with the topology of uniform convergence on bounded subsets of X. If X is a bornological space and if Y is complete then Lb(X; Y) is a complete TVS.[3]
    • In particular, the strong dual of a locally convex bornological space is complete.[3] However, it need not be bornological.
Subsets
  • In a locally convex bornological space, every convex bornivorous set B is a neighborhood of 0 (B is not required to be a disk).[3]
  • Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[3]
  • Closed vector subspaces of bornological space need not be bornological.[3]

Ultrabornological spaces
Main article: Ultrabornological space

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 equivalent conditions hold:

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

References
  1. Narici & Beckenstein 2011, pp. 156-175.
  2. Wilansky 2013, p. 50.
  3. Narici & Beckenstein 2011, pp. 441-457.
  4. Swartz 1992, pp. 15-16.
  5. Narici & Beckenstein 2011, pp. 453-454.
  6. Adasch, Ernst & Keim 1978, pp. 60-61.
  7. Wilansky 2013, p. 48.
  8. Narici & Beckenstein 2011, p. 450.
  9. Adasch, Ernst & Keim 1978, pp. 60-65.
  10. Narici & Beckenstein 2011, p. 453.
  11. Schaefer & Wolff 1999, p. 144.
  12. Khaleelulla 1982, pp. 28-63.
  13. Schaefer & Wolff 1999, pp. 103-110.
  14. Adasch, Ernst & Keim 1978, pp. 70-73.

Bibliography
  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 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. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
  • Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
  • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 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.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
  • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.