Cayley table

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

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 called bounded sets, and 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 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

If $B_1$ and $B_2$ are two bornologies over the spaces $X$ and $Y$, respectively, and if $f:X\to Y$ 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 $f:X\to Y$ 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,
  • utakes bounded disks to bounded disks,
  • For every bornivorous disk D in Y, $u^{-1}(D)$ is 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 $A$.

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 $X^{\prime }$ 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 operators 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 $\tau (X,X^{\prime })$ 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.
• Frechet 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 $X^{\prime }$. Then the following are equivalent:
  • $\beta (X^{\prime },X)$ is bornological,
  • $\beta (X^{\prime },X)$ is quasi-barrelled,
  • $\beta (X^{\prime },X)$ is barrelled,
  • $X$ is a distingushed 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 $B\subset X$ that's bounded in $X$, $u(B)$ is bounded,
  • If $(x_n)\subset X$ 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 we have that ${\mu }_D$ is a norm and so that $X_D$ is a normed space. If D is a bounded sequentially complete disk andX is Hausdorff then the space $X_D$ is in fact a Banach space. And 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 $B\subseteq rT$.

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 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 $D^{\prime }$ be the polar of U then $D^{\prime }$ 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

References

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