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.

Bornological sets

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, ..., B_n ∈ 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 (X, B) 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

• 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 $P_\kappa(X)$ of all subsets of $X$ having cardinality at most $\kappa$ 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

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

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

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 $U\subseteq X$containing 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 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 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 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′).
  • 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 $X'$. 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 $u : X \to Y$ 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

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

• 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 $X_D$ 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

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

See also

• Space of linear maps