# 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 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

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

• 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.

Examples:

Theorems:

## 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

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:

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.

## 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

## 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:

### Properties

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