# 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 after *borné*, the French word for "bounded".

## Contents

## Bornological sets[edit]

A **bornology** on a set *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**. 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[edit]

- 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 of all subsets of*X*having cardinality at most is a bornology when is an infinite cardinal. - For any topological space
*X*that is T_{1}, the set of subsets of*X*with compact closure is a bornology.

## Bounded maps[edit]

If *B _{1}* and

*B*are two bornologies over the spaces

_{2}*X*and

*Y*, respectively, and if

*f : X → Y*is a function, then we say that

*f*is a

**bounded map**if it maps

*B*-bounded sets in

_{1}*X*to

*B*-bounded sets in

_{2}*Y*. If in addition

*f*is a bijection and 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 → 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 → 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*, is also bornivorous.

## Vector bornologies[edit]

If *X* is a vector space over a field *K* 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 ) then it is called

**separated**. A subset

*A*of

*X*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[edit]

Every topological vector space *X* gives a bornology on *X* by defining a subset *B ⊆ X* to be bounded (or von-Neumann bounded), if and only if for all open sets *U ⊆ X* containing zero there exists a *r > 0* with *B ⊆ r U*. If *X* is a locally convex topological vector space then *B ⊆ 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[edit]

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[edit]

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 topology and 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
*X*is the same as ,*X'*s given topology. - Every convex, balanced, and bornivorous set in
*X*is a neighborhood of zero. - Every bounded semi-norm on
*X*is continuous, - Any other Hausdorff locally convex topological vector space topology on
*X*that has the same (von-Neumann) bornology as is necessarily coarser than . - 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*as_{D}*D*varies over the closed and bounded disks of*X*(or as*D*varies over the bounded disks of*X*).*X*carries the Mackey topology 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[edit]

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[edit]

- 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 . Then the following are equivalent:- is bornological,
- is quasi-barrelled,
- is barrelled,
*X*is a distinguished space.

- If
*X*is bornological, is a locally convex TVS, and*u : X → Y*is a linear map, then the following are equivalent:*u*is continuous,- for every set
*B ⊆ X*that's bounded in*X*,*u(B)*is bounded, - If
*(x*is a null sequence in_{n}) ⊆ X*X*then*(u(x*is a null sequence in_{n}))*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[edit]

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 *p _{D}*. 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. If

*D*is Von-Neuman bounded in

*X*then the (normed) topology of

*X*will be finer than the subspace topology that

_{D}*X*induces on this set.

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 *X _{D}* is a normed space. If

*D*is a bounded sequentially complete disk and

*X*is Hausdorff, then the space

*X*is a Banach space. A bounded disk in

_{D}*X*for which

*X*is a Banach space is called a

_{D}**Banach disk**,

**infracomplete**, or a

**bounded completant**.

### Properties[edit]

Suppose that *X* is a locally convex Hausdorff space. If *D* is a bounded Banach disk in *X* and *T* is a barrel in *X* then *T* absorbs *D* (i.e. there is a number *r > 0* such that *D ⊆ r T*).

### Examples[edit]

- 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[edit]

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*as_{D}*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 → Y*, if*u*is bounded on each Banach disk then*u*is continuous. - For every Banach space
*Y*and every linear map*u : X → Y*, if*u*is bounded on each Banach disk then*u*is continuous.

### Properties[edit]

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

## See also[edit]

## References[edit]

- Hogbe-Nlend, Henri (1977).
*Bornologies and functional analysis*. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064. - H.H. Schaefer (1970).
*Topological Vector Spaces*. GTM.**3**. Springer-Verlag. pp. 61–63. ISBN 0-387-05380-8. - Khaleelulla, S.M. (1982).
*Counterexamples in Topological Vector Spaces*. GTM.**936**. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656. - Kriegl, Andreas; Michor, Peter W. (1997).
*The Convenient Setting of Global Analysis*. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.