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 functions, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. 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 such that

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

If in addition ℬ is stable under the formation of convex hulls (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 X is the trivial subspace (i.e. the space consisting only of 0) then it is called separated.

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]

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 X is called the bornology or the Von-Neumann bornology of X.

Induced topology

Suppose that we start with a vector space X and convex vector bornology B on X. 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 X that is compatible with the vector space structure of X.

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.

Note that if X is a TVS in which every bornivorous set is a neighborhood of the origin, then any bounded linear map from X into any other TVS is continuous.[3]

Definition: A X with topology 𝜏 and continuous dual X ' is called a bornological space if any of the following equivalent conditions holds:

If X is Hausdorff:

  1. Every bounded linear operator from X into another TVS is continuous.[4]
  2. Every bounded linear operator from X into a complete metrizable TVS is continuous.[4]

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

    The locally convex topology induced by the von-Neumann bornology on X is the same as 𝜏, X's given topology.
  1. Every convex, balanced, and bornivorous set in X is a neighborhood of zero.
  2. Every bounded semi-norm on X is continuous.
  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. For all locally convex spaces Y, every bounded linear operator from X into Y is continuous.
  5. X is the inductive limit of normed spaces.
  6. 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).
  7. X carries the Mackey topology and all bounded linear functionals on X are continuous.
  8. 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.

Examples

The following topological vector spaces are all bornological:

  • Any metrizable TVS is bornological.[5] In particular, any Fréchet space is bornological.
  • Any LF-space (i.e. any locally convex space that is the strict inductive limit of Fréchet spaces).
  • A countable product of Hausdorff bornological spaces is bornological.[5]
  • Quotients of Hausdorff bornological spaces are bornological.[5]
  • The direct sum and inductive limit of Hausdorff bornological spaces is bornological.[5]
  • Fréchet Montel spaces have a bornological strong dual.
  • The strong dual of every reflexive Fréchet space is bornological.[6]
  • If the strong dual of a metrizable locally convex space is separable, then it is bornological.[6]
  • A vector subspace of a Hausdorff bornological space X that has finite codimension in X is bornological.[5]

Counter-examples
  • There exists a bornological LB-space whose strong bidual is not bornological.[7]
  • A closed vector subspace of a bornological space is not necessarily bornological.[8]

Properties
  • Every Hausdorff bornological space is quasi-barrelled.[9]
  • Given a bornological space X with continuous dual X′, then 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:
    • is bornological,
    • is quasi-barrelled,
    • is barrelled,
    • X is a distinguished space.
  • If X is bornological, Y is a locally convex TVS, and u : X → Y is a linear map, then the following are equivalent:
    • u is continuous,
    • for every set B ⊆ X that's bounded in X, u(B) is bounded,
    • If (xn) ⊆ X is a null sequence in X then (u(xn)) is a null sequence in Y.
  • The strong dual of a bornological space is complete, but it need not be bornological.
  • Closed subspaces of bornological space need not be bornological.

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 conditions hold:

  • every infrabornivorous disk is a neighborhood of 0,
  • X be the inductive limit of the spaces XD as D varies over all compact disks in X,
  • A seminorm on X that is bounded on each Banach disk is necessarily continuous,
  • 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.
  • 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. Wilansky 2013, p. 48.
  4. Adasch 1978, pp. 60-61.
  5. Adasch 1978, pp. 60-65.
  6. Schaefer & Wolff 1999, p. 144.
  7. Khaleelulla 1982, pp. 28-63.
  8. Schaefer & Wolff 1999, pp. 103-110.
  9. Adasch 1978, pp. 70-73.

Bibliography
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