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.
A bornology on a set X is a collection B of subsets of X such that
- B covers X, i.e.
- 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
Elements of the collection B are usually called bounded sets. The pair (X, B) 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.
- 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 of all subsets of X having cardinality at most is a bornology when is an infinite cardinal.
- For any topological space X that is T1, the set of subsets of X with compact closure is a bornology.
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 is also bounded then we say that f is a bornological isomorphism.
- 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.
- Suppose that X and Y are locally convex spaces and that u : X → Yis 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, is also bornivorous.
If X is a vector space over a field K and 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 ) then it is called separated. A subset A of B 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.
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.
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 and continuous dual 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 Xis the same as , 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 is necessarily coarser than .
- 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 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.
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.
- Given a bornological space X with continuous dual X′, then 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 Xbe a metrizable locally convex space with continuous dual . Then the following are equivalent:
- If X is bornological, 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.
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 or by pD. When we give span(D) the topology induced by this seminorm, we denote the resulting topological vector space by . 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 and D be the x-axis. However, if D is a bounded disk and if X is Hausdorff, then 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.
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).
- 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 is denoted by so that is a complete Hausdorff space and is a norm on this space making into a Banach space. The polar of U, , is a weakly compact bounded equicontinuous disk in and so is infracomplete.
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.
- The finite product of ultrabornological spaces is ultrabornological.
- Inductive limits of ultrabornological spaces are ultrabornological.
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