Bornological space

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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 and their name was given by Bourbaki.

Bornological sets[edit]

Let X be any set. A bornology on 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. However, if it is necessary to differentiate this formal usage of the term "bounded" with traditional uses, elements of the collection B may also be called bornivorous sets. The pair (XB) is called a bornological set.

A base of the bornology B is a subset B_0 of B such that each element of B is a subset of an element of B_0.

Examples[edit]

  • For any set X, the discrete topology of X is a bornology.
  • For any set X, the set of finite (or countably infinite) subsets of X is a bornology.
  • For any topological space X that is T1, the set of subsets of X with compact closure is a bornology.

Bounded maps[edit]

If B_1 and B_2 are two bornologies over the spaces X and Y, respectively, and if f\colon X \rightarrow Y is a function, then we say that f is a bounded map if it maps B_1-bounded sets in X to B_2-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\colon X \rightarrow 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 \to 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[edit]

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 0) 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[edit]

Every topological vector space X gives a bornology on X by defining a subset B\subseteq X to be bounded (or von-Neumann bounded), if and only if for all open sets U\subseteq Xcontaining zero there exists a \lambda>0 with B\subseteq\lambda U. If X is a locally convex topological vector space then B\subseteq X is bounded if and only if all continuous semi-norms on X are bounded on A.

The set of all bounded subsets of X is called the bornology or the Von-Neumann bornology of X.

Induced topology[edit]

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[edit]

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 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 X's initial topology,
  • Every bounded semi-norm on X is continuous,
  • 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 X_D as D varies over the closed and bounded disks of X (or as D varies over the bounded disks of X).
  • Every convex, balanced, and bornivorous set in X is a neighborhood of 0.
  • X caries 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[edit]

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.
  • Frechet Montel have a bornological strong dual.

Properties[edit]

  • 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 X'. Then the following are equivalent:
  • If X is bornological, Y is a locally convex TVS, and u : X \to Y is a linear map, then the following are equivalent:
    • u is continuous,
    • for every set  B \sub X that's bounded in X, u(B) is bounded,
    • If (x_n) \sub X is a null sequence in X then (u(x_n)) 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[edit]

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 p_D. 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.

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 X_D is a normed space. If D is a bounded sequentially complete disk and X is Hausdorff, then the space X_D is a Banach space. A bounded disk in X for which X_D is a Banach space is called a Banach disk, infracomplete, or a bounded completant.

Suppose that X is a locally convex Hausdorff space and that D is a bounded disk in X. Then

  • If D is complete in X and T is a Barrell in X, then there is a number r > 0 such that B \subseteq r T.

Examples[edit]

  • 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 we can place on X the topological vector space topology induced by the neighborhoods r U where r > 0 ranges over the positive real numbers. When X has this topology, it is denoted by X_U. However, this topology is not necessarily Hausdorff or complete so we denote the completion of the Hausdorff space X_U/\ker(\mu_U) by \hat{X}_U so that \hat{X}_U is a complete Hausdorff space and \mu_U is a norm on this space so that \hat{X}_U is a Banach space. If we let D' be the polar of U, then D' is a weakly compact bounded equicontinuous disk in X^* and so is infracomplete.

Ultrabornological spaces[edit]

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 X_D 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 \to Y, if u is bounded on each Banach disk then u is continuous.
  • For every Banach space Y and every linear map u : X \to Y, if u is bounded on each Banach disk then u is continuous.

Properties[edit]

  • The finite product of ultrabornological spaces is ultrabornological.
  • Inductive limits of ultrabornological spaces are ultrabornological.

See also[edit]

References[edit]