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".

Bornological sets

A bornology on a set X is a collection B of subsets of X such that

• B covers X, i.e. $X=\bigcup B;$ • 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 $\bigcup _{i=1}^{n}B_{i}\in B.$ Elements of the collection B are usually called bounded sets. The pair (XB) is called 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.

Examples

• For any set X, the power set of X is a bornology.
• For any set X, the set of finite subsets of X is a bornology. Similarly the set of all at most countably infinite subsets is a bornology. More generally: The set $\wp _{_{\kappa }}(X)$ of all subsets of X having cardinality at most $\kappa$ is a bornology when $\kappa$ is an infinite cardinal.
• For any topological space X that is T1, the set of subsets of X with compact closure is a bornology.

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 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.

Examples:

• If X and Y are any two topological vector spaces (they need not even be Hausdorff) and if f : X → Y is a continuous linear operator between them, then f is a bounded linear operator (when X and Y have their von-Neumann bornologies). The converse is in general false.

Theorems:

• Suppose that X and Y are locally convex spaces and that u : X → Y is a linear map. Then the following are equivalent:
• u is a bounded map,
• u takes bounded disks to bounded disks,
• For every bornivorous (i.e. bounded in the bornological sense) disk D in Y, $u^{-1}(D)$ is also bornivorous.

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 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 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.

Bornology of a topological vector space

Every topological vector space X 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 ⊆ r U. 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 bornological space is a locally convex topological vector space whose topology can be recovered from its bornology in a natural way. Explicitly, a Hausdorff locally convex space X with topology $\tau$ and continuous dual $X'$ is called a bornological space if any one of the following equivalent conditions holds:

• The locally convex topology induced by the von-Neumann bornology on X is the same as $\tau$ , X's given topology.
• Every convex, balanced, and bornivorous set in X is a neighborhood of zero.
• Every bounded semi-norm on X is continuous,
• Any other Hausdorff locally convex topological vector space topology on X that has the same (von-Neumann) bornology as $(X,\tau )$ is necessarily coarser than $\tau$ .
• For all locally convex spaces Y, every bounded linear operator from X into Y is continuous.
• X is the inductive limit of normed spaces.
• 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).
• X carries the Mackey topology $\tau (X,X')$ and all bounded linear functionals on X are continuous.
• 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 metrisable locally convex space is bornological. In particular, any Fréchet space.
• Any LF-space (i.e. any locally convex space that is the strict inductive limit of Fréchet spaces).
• Separated quotients of bornological spaces are bornological.
• The locally convex direct sum and inductive limit of bornological spaces is bornological.
• Fréchet Montel have a bornological strong dual.

Properties

• 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 Xbe a metrizable locally convex space with continuous dual $X'$ . Then the following are equivalent:
• $\beta (X',X)$ is bornological,
• $\beta (X',X)$ is quasi-barrelled,
• $\beta (X',X)$ 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.

Banach disks

Suppose that X is a topological vector space. Then we say that a subset D of X is a disk if it is convex and balanced. The disk D is absorbing in the space span(D) and so its Minkowski functional forms a seminorm on this space, which is denoted by $\mu _{D}$ or by pD. When we give span(D) the topology induced by this seminorm, we denote the resulting topological vector space by $X_{D}$ . A basis of neighborhoods of 0 of this space consists of all sets of the form r D where r ranges over all positive real numbers. If D is Von-Neuman bounded in X then the (normed) topology of XD will be finer than the subspace topology that X induces on this set.

This space is not necessarily Hausdorff as is the case, for instance, if we let $X=\mathbb {R} ^{2}$ and D be the x-axis. However, if D is a bounded disk and if X is Hausdorff, then $\mu _{D}$ is a norm and XD is a normed space. If D is a bounded sequentially complete disk and X is Hausdorff, then the space XD is a Banach space. A bounded disk in X for which XD is a Banach space is called a Banach disk, infracomplete, or a bounded completant.

Properties

Suppose that X is a locally convex Hausdorff space. If D is a bounded Banach disk in X and T is a barrel in X then T absorbs D (i.e. there is a number r > 0 such that D ⊆ r T).

Examples

• Any closed and bounded disk in a Banach space is a Banach disk.
• If U is a convex balanced closed neighborhood of 0 in X then the collection of all neighborhoods r U, where r > 0 ranges over the positive real numbers, induces a topological vector space topology on X. When X has this topology, it is denoted by X_U. Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space $X_{U}/\ker(\mu _{U})$ is denoted by ${\hat {X}}_{U}$ so that ${\hat {X}}_{U}$ is a complete Hausdorff space and $\mu _{U}$ is a norm on this space making ${\hat {X}}_{U}$ into a Banach space. The polar of U, $D'$ , is a weakly compact bounded equicontinuous disk in $X^{*}$ and so is infracomplete.

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 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.