# 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 ${\displaystyle B_{0}}$ of B such that each element of B is a subset of an element of ${\displaystyle B_{0}}$.

### 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 ${\displaystyle B_{1}}$ and ${\displaystyle B_{2}}$ are two bornologies over the spaces ${\displaystyle X}$ and ${\displaystyle Y}$, respectively, and if ${\displaystyle f\colon X\rightarrow Y}$ is a function, then we say that ${\displaystyle f}$ is a bounded map if it maps ${\displaystyle B_{1}}$-bounded sets in ${\displaystyle X}$ to ${\displaystyle B_{2}}$-bounded sets in ${\displaystyle Y}$. If in addition ${\displaystyle f}$ is a bijection and ${\displaystyle f^{-1}}$ is also bounded then we say that ${\displaystyle f}$ is a bornological isomorphism.

Examples:

Theorems:

## Vector bornologies

If ${\displaystyle X}$ is a vector space over a field K and then a vector bornology on ${\displaystyle X}$ is a bornology B on ${\displaystyle X}$ 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 ${\displaystyle X}$ is the trivial subspace (i.e. the space consisting only of ${\displaystyle 0}$) 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 ${\displaystyle B\subseteq X}$ to be bounded (or von-Neumann bounded), if and only if for all open sets ${\displaystyle U\subseteq X}$containing zero there exists a ${\displaystyle \lambda >0}$ with ${\displaystyle B\subseteq \lambda U}$. If X is a locally convex topological vector space then ${\displaystyle B\subseteq X}$ 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 ${\displaystyle X}$ and convex vector bornology B on ${\displaystyle X}$. 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 ${\displaystyle X}$ that is compatible with the vector space structure of ${\displaystyle X}$.

## 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 ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ is called a bornological space if any one of the following equivalent conditions holds:

where a subset A of ${\displaystyle X}$ 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 ${\displaystyle \mu _{D}}$ or by ${\displaystyle p_{D}}$. When we give span(D) the topology induced by this seminorm, we denote the resulting topological vector space by ${\displaystyle X_{D}}$. 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 ${\displaystyle X=\mathbb {R} ^{2}}$ and D be the x-axis. However, if D is a bounded disk and if X is Hausdorff, then ${\displaystyle \mu _{D}}$ is a norm and ${\displaystyle X_{D}}$ is a normed space. If D is a bounded sequentially complete disk and X is Hausdorff, then the space ${\displaystyle X_{D}}$ is a Banach space. A bounded disk in X for which ${\displaystyle X_{D}}$ 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.