# 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

*B*covers*X*, i.e.*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*B*_{1}, ...,*B*_{n}∈*B*, then

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 of *B* such that each element of *B* is a subset of an element of .

### 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 and are two bornologies over the spaces and , respectively, and if is a function, then we say that is a **bounded map** if it maps -bounded sets in to -bounded sets in . If in addition is a bijection and is also bounded then we say that is a **bornological isomorphism**.

Examples:

- If and are any two topological vector spaces (they need not even be Hausdorff) and if is a continuous linear operator between them, then is a bounded linear operator (when and have their von-Neumann bornologies). The converse is in general false.

Theorems:

- Suppose that
*X*and*Y*are locally convex spaces and that is a linear map. Then the following are equivalent:

## Vector bornologies

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

## 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 with continuous dual 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 is the same as 's initial topology,
- Every bounded semi-norm on is continuous,
- For all locally convex spaces
*Y*, every bounded linear operator from into 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 is a neighborhood of .
*X*caries the Mackey topology and all bounded linear functionals on*X*are continuous.- has both of the following properties:

where a subset *A* of 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 be a metrizable locally convex space with continuous dual . Then the following are equivalent:
- is bornological,
- is quasi-barrelled,
- is barrelled,
- is a distinguished space.

- If is bornological, is a locally convex TVS, and is a linear map, then the following are equivalent:
- 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 or by . When we give *span(D)* the topology induced by this seminorm, we denote the resulting topological vector space by . 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 and *D* be the *x*-axis. However, if *D* is a bounded disk and if *X* is Hausdorff, then is a norm and is a normed space. If *D* is a bounded sequentially complete disk and *X* is Hausdorff, then the space is a Banach space. A bounded disk in *X* for which 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

### 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 is denoted by so that is a complete Hausdorff space and is a norm on this space making into a Banach space. The polar of*U*, , is a weakly compact bounded equicontinuous disk in 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 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 , if*u*is bounded on each Banach disk then*u*is continuous. - For every Banach space
*Y*and every linear map , 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

## References

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