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

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, ..., B_n ∈ ℬ, then B₁ ∪ ⋅⋅⋅ ∪ B_n ∈ 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

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_• = (x_i)^∞
_i=1 in a TVS X is said to be Mackey convergent to 0 if there exists a sequence of positive real numbers r_• = (r_i)^∞
_i=1 diverging to ∞ such that (r_ix_i)^∞
_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 𝒩_ℬ(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 X_b denote X endowed with the topology induced by von Neumann bornology of X. Define Y_b similarly. Then a linear map L : X → Y is a bounded linear operator if and only if L : X_b → Y is continuous.

Moreover, if X is bornological, Y is Hausdorff, and L : X → Y is continuous linear map then so is L : X → Y_b. If in addition X is also ultrabornological, then the continuity of L : X → Y implies the continuity of L : X → Y_ub, where Y_ub 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]

5. The locally convex topology induced by the von Neumann bornology on X is the same as τ, X's given topology.
6. Every bounded seminorm on X is continuous.[3]
7. 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.
8. X is the inductive limit of normed spaces.[3]
9. 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).[3]
10. X carries the Mackey topology ${\displaystyle \tau (X,X')}$ and all bounded linear functionals on X are continuous.[3]
11. 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_• = (X_i)_{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:

• Any locally convex pseudometrizable TVS is bornological.[3][9]
  • Thus every normed space and Fréchet space is bornological.
• 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.[10][9]
• Quotients of Hausdorff locally convex bornological spaces are bornological.[9]
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.[9]
• Fréchet Montel spaces have bornological strong duals.
• The strong dual of every reflexive Fréchet space is bornological.[11]
• If the strong dual of a metrizable locally convex space is separable, then it is bornological.[11]
• A vector subspace of a Hausdorff locally convex bornological space X that has finite codimension in X is bornological.[3][9]
• The finest locally convex topology on a vector space is bornological.[3]

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′).
  • In particular, bornological spaces are Mackey spaces.
• 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 ${\displaystyle X'}$. Then the following are equivalent:
  1. ${\displaystyle \beta (X',X)}$ is bornological.
  2. ${\displaystyle \beta (X',X)}$ is quasi-barrelled.
  3. ${\displaystyle \beta (X',X)}$ 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 (x_n) ⊆ X is a null sequence in X then (L(x_n)) is a null sequence in Y.
  5. If (x_n) ⊆ X is a Mackey convergent null sequence in X then (L(x_n)) is a bounded subset of Y.
• Suppose that X and Y are locally convex TVSs and that the space of continuous linear maps L_b(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 L_b(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

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 X_D 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

• Bornology
• Bornivorous set – A set that can absorb any bounded subset
• Bounded set (topological vector space)
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Space of linear maps
• Topological vector space – Vector space with a notion of nearness
• Vector bornology

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.