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Bornological space

## Summary

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 linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

Bornological spaces were first studied by George Mackey.[citation needed] The name was coined by Bourbaki[citation needed] after borné, the French word for "bounded".

## Bornologies and bounded maps

A bornology on a set ${\displaystyle X}$  is a collection ${\displaystyle {\mathcal {B}}}$  of subsets of ${\displaystyle X}$  that satisfy all the following conditions:

1. ${\displaystyle {\mathcal {B}}}$  covers ${\displaystyle X;}$  that is, ${\displaystyle X=\cup {\mathcal {B}}}$ ;
2. ${\displaystyle {\mathcal {B}}}$  is stable under inclusions; that is, if ${\displaystyle B\in {\mathcal {B}}}$  and ${\displaystyle A\subseteq B,}$  then ${\displaystyle A\in {\mathcal {B}}}$ ;
3. ${\displaystyle {\mathcal {B}}}$  is stable under finite unions; that is, if ${\displaystyle B_{1},\ldots ,B_{n}\in {\mathcal {B}}}$  then ${\displaystyle B_{1}\cup \cdots \cup B_{n}\in {\mathcal {B}}}$ ;

Elements of the collection ${\displaystyle {\mathcal {B}}}$  are called ${\displaystyle {\mathcal {B}}}$ -bounded or simply bounded sets if ${\displaystyle {\mathcal {B}}}$  is understood.[1] The pair ${\displaystyle (X,{\mathcal {B}})}$  is called a bounded structure or a bornological set.[1]

A base or fundamental system of a bornology ${\displaystyle {\mathcal {B}}}$  is a subset ${\displaystyle {\mathcal {B}}_{0}}$  of ${\displaystyle {\mathcal {B}}}$  such that each element of ${\displaystyle {\mathcal {B}}}$  is a subset of some element of ${\displaystyle {\mathcal {B}}_{0}.}$  Given a collection ${\displaystyle {\mathcal {S}}}$  of subsets of ${\displaystyle X,}$  the smallest bornology containing ${\displaystyle {\mathcal {S}}}$  is called the bornology generated by ${\displaystyle {\mathcal {S}}.}$ [2]

If ${\displaystyle (X,{\mathcal {B}})}$  and ${\displaystyle (Y,{\mathcal {C}})}$  are bornological sets then their product bornology on ${\displaystyle X\times Y}$  is the bornology having as a base the collection of all sets of the form ${\displaystyle B\times C,}$  where ${\displaystyle B\in {\mathcal {B}}}$  and ${\displaystyle C\in {\mathcal {C}}.}$ [2] A subset of ${\displaystyle X\times Y}$  is bounded in the product bornology if and only if its image under the canonical projections onto ${\displaystyle X}$  and ${\displaystyle Y}$  are both bounded.

### Bounded maps

If ${\displaystyle (X,{\mathcal {B}})}$  and ${\displaystyle (Y,{\mathcal {C}})}$  are bornological sets then a function ${\displaystyle f:X\to Y}$  is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps ${\displaystyle {\mathcal {B}}}$ -bounded subsets of ${\displaystyle X}$  to ${\displaystyle {\mathcal {C}}}$ -bounded subsets of ${\displaystyle Y;}$  that is, if ${\displaystyle f({\mathcal {B}})\subseteq {\mathcal {C}}.}$ [2] If in addition ${\displaystyle f}$  is a bijection and ${\displaystyle f^{-1}}$  is also bounded then ${\displaystyle f}$  is called a bornological isomorphism.

## Vector bornologies

Let ${\displaystyle X}$  be a vector space over a field ${\displaystyle \mathbb {K} }$  where ${\displaystyle \mathbb {K} }$  has a bornology ${\displaystyle {\mathcal {B}}_{\mathbb {K} }.}$  A bornology ${\displaystyle {\mathcal {B}}}$  on ${\displaystyle X}$  is called a vector bornology on ${\displaystyle X}$  if it 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 ${\displaystyle X}$  is a topological vector space (TVS) and ${\displaystyle {\mathcal {B}}}$  is a bornology on ${\displaystyle X,}$  then the following are equivalent:

1. ${\displaystyle {\mathcal {B}}}$  is a vector bornology;
2. Finite sums and balanced hulls of ${\displaystyle {\mathcal {B}}}$ -bounded sets are ${\displaystyle {\mathcal {B}}}$ -bounded;[2]
3. The scalar multiplication map ${\displaystyle \mathbb {K} \times X\to X}$  defined by ${\displaystyle (s,x)\mapsto sx}$  and the addition map ${\displaystyle X\times X\to X}$  defined by ${\displaystyle (x,y)\mapsto x+y,}$  are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).[2]

A vector bornology ${\displaystyle {\mathcal {B}}}$  is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then ${\displaystyle {\mathcal {B}}.}$  And a vector bornology ${\displaystyle {\mathcal {B}}}$  is called separated if the only bounded vector subspace of ${\displaystyle X}$  is the 0-dimensional trivial space ${\displaystyle \{0\}.}$

Usually, ${\displaystyle \mathbb {K} }$  is either the real or complex numbers, in which case a vector bornology ${\displaystyle {\mathcal {B}}}$  on ${\displaystyle X}$  will be called a convex vector bornology if ${\displaystyle {\mathcal {B}}}$  has a base consisting of convex sets.

### Bornivorous subsets

A subset ${\displaystyle A}$  of ${\displaystyle X}$  is called bornivorous and a bornivore if it absorbs every bounded set.

In a vector bornology, ${\displaystyle A}$  is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology ${\displaystyle A}$  is bornivorous if it absorbs every bounded disk.

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.[3]

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[4]

#### Mackey convergence

A sequence ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$  in a TVS ${\displaystyle X}$  is said to be Mackey convergent to ${\displaystyle 0}$  if there exists a sequence of positive real numbers ${\displaystyle r_{\bullet }=(r_{i})_{i=1}^{\infty }}$  diverging to ${\displaystyle \infty }$  such that ${\displaystyle (r_{i}x_{i})_{i=1}^{\infty }}$  converges to ${\displaystyle 0}$  in ${\displaystyle X.}$ [5]

### Bornology of a topological vector space

Every topological vector space ${\displaystyle X,}$  at least on a non discrete valued field gives a bornology on ${\displaystyle X}$  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 ${\displaystyle r>0}$  with ${\displaystyle B\subseteq rU.}$  If ${\displaystyle X}$  is a locally convex topological vector space then ${\displaystyle B\subseteq X}$  is bounded if and only if all continuous semi-norms on ${\displaystyle X}$  are bounded on ${\displaystyle B.}$

The set of all bounded subsets of a topological vector space ${\displaystyle X}$  is called the bornology or the von Neumann bornology of ${\displaystyle X.}$

If ${\displaystyle X}$  is a locally convex topological vector space, then an absorbing disk ${\displaystyle D}$  in ${\displaystyle X}$  is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).[4]

### Induced topology

If ${\displaystyle {\mathcal {B}}}$  is a convex vector bornology on a vector space ${\displaystyle X,}$  then the collection ${\displaystyle {\mathcal {N}}_{\mathcal {B}}(0)}$  of all convex balanced subsets of ${\displaystyle X}$  that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on ${\displaystyle X}$  called the topology induced by ${\displaystyle {\mathcal {B}}}$ .[4]

If ${\displaystyle (X,\tau )}$  is a TVS then the bornological space associated with ${\displaystyle X}$  is the vector space ${\displaystyle X}$  endowed with the locally convex topology induced by the von Neumann bornology of ${\displaystyle (X,\tau ).}$ [4]

Theorem[4] — Let ${\displaystyle X}$  and ${\displaystyle Y}$  be locally convex TVS and let ${\displaystyle X_{b}}$  denote ${\displaystyle X}$  endowed with the topology induced by von Neumann bornology of ${\displaystyle X.}$  Define ${\displaystyle Y_{b}}$  similarly. Then a linear map ${\displaystyle L:X\to Y}$  is a bounded linear operator if and only if ${\displaystyle L:X_{b}\to Y}$  is continuous.

Moreover, if ${\displaystyle X}$  is bornological, ${\displaystyle Y}$  is Hausdorff, and ${\displaystyle L:X\to Y}$  is continuous linear map then so is ${\displaystyle L:X\to Y_{b}.}$  If in addition ${\displaystyle X}$  is also ultrabornological, then the continuity of ${\displaystyle L:X\to Y}$  implies the continuity of ${\displaystyle L:X\to Y_{ub},}$  where ${\displaystyle Y_{ub}}$  is the ultrabornological space associated with ${\displaystyle Y.}$

## Quasi-bornological spaces

Quasi-bornological spaces where introduced by S. Iyahen in 1968.[6]

A topological vector space (TVS) ${\displaystyle (X,\tau )}$  with a continuous dual ${\displaystyle X^{\prime }}$  is called a quasi-bornological space[6] if any of the following equivalent conditions holds:

1. Every bounded linear operator from ${\displaystyle X}$  into another TVS is continuous.[6]
2. Every bounded linear operator from ${\displaystyle X}$  into a complete metrizable TVS is continuous.[6][7]
3. Every knot in a bornivorous string is a neighborhood of the origin.[6]

Every pseudometrizable TVS is quasi-bornological. [6] A TVS ${\displaystyle (X,\tau )}$  in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.[8] If ${\displaystyle X}$  is a quasi-bornological TVS then the finest locally convex topology on ${\displaystyle X}$  that is coarser than ${\displaystyle \tau }$  makes ${\displaystyle X}$  into a locally convex bornological space.

### Bornological space

In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.

Every locally convex quasi-bornological space is bornological but there exist bornological spaces that are not quasi-bornological.[6]

A topological vector space (TVS) ${\displaystyle (X,\tau )}$  with a continuous dual ${\displaystyle X^{\prime }}$  is called a bornological space if it is locally convex and any of the following equivalent conditions holds:

1. Every convex, balanced, and bornivorous set in ${\displaystyle X}$  is a neighborhood of zero.[4]
2. Every bounded linear operator from ${\displaystyle X}$  into a locally convex TVS is continuous.[4]
• Recall that a linear map is bounded if and only if it maps any sequence converging to ${\displaystyle 0}$  in the domain to a bounded subset of the codomain.[4] In particular, any linear map that is sequentially continuous at the origin is bounded.
3. Every bounded linear operator from ${\displaystyle X}$  into a seminormed space is continuous.[4]
4. Every bounded linear operator from ${\displaystyle X}$  into a Banach space is continuous.[4]

If ${\displaystyle X}$  is a Hausdorff locally convex space then we may add to this list:[7]

1. The locally convex topology induced by the von Neumann bornology on ${\displaystyle X}$  is the same as ${\displaystyle \tau ,}$  ${\displaystyle X}$ 's given topology.
2. Every bounded seminorm on ${\displaystyle X}$  is continuous.[4]
3. Any other Hausdorff locally convex topological vector space topology on ${\displaystyle X}$  that has the same (von Neumann) bornology as ${\displaystyle (X,\tau )}$  is necessarily coarser than ${\displaystyle \tau .}$
4. ${\displaystyle X}$  is the inductive limit of normed spaces.[4]
5. ${\displaystyle X}$  is the inductive limit of the normed spaces ${\displaystyle X_{D}}$  as ${\displaystyle D}$  varies over the closed and bounded disks of ${\displaystyle X}$  (or as ${\displaystyle D}$  varies over the bounded disks of ${\displaystyle X}$ ).[4]
6. ${\displaystyle X}$  carries the Mackey topology ${\displaystyle \tau (X,X^{\prime })}$  and all bounded linear functionals on ${\displaystyle X}$  are continuous.[4]
7. ${\displaystyle X}$  has both of the following properties:
• ${\displaystyle X}$  is convex-sequential or C-sequential, which means that every convex sequentially open subset of ${\displaystyle X}$  is open,
• ${\displaystyle X}$  is sequentially bornological or S-bornological, which means that every convex and bornivorous subset of ${\displaystyle X}$  is sequentially open.
where a subset ${\displaystyle A}$  of ${\displaystyle X}$  is called sequentially open if every sequence converging to ${\displaystyle 0}$  eventually belongs to ${\displaystyle A.}$

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,[4] where recall that a linear operator is sequentially continuous if and only if it is sequentially continuous at the origin. Thus for linear maps from a bornological space into a locally convex space, continuity is equivalent to sequential continuity at the origin. More generally, we even have the following:

• Any linear map ${\displaystyle F:X\to Y}$  from a locally convex bornological space into a locally convex space ${\displaystyle Y}$  that maps null sequences in ${\displaystyle X}$  to bounded subsets of ${\displaystyle Y}$  is necessarily continuous.

### Sufficient conditions

Mackey–Ulam theorem[9] — The product of a collection ${\displaystyle X_{\bullet }=(X_{i})_{i\in I}}$  locally convex bornological spaces is bornological if and only if ${\displaystyle I}$  does not admit an Ulam measure.

As a consequent of the Mackey–Ulam theorem, "for all practical purposes, the product of bornological spaces is bornological."[9]

The following topological vector spaces are all bornological:

• Any locally convex pseudometrizable TVS is bornological.[4][10]
• Any strict inductive limit of bornological spaces, in particular any strict LF-space, is bornological.
• This shows that there are bornological spaces that are not metrizable.
• A countable product of locally convex bornological spaces is bornological.[11][10]
• Quotients of Hausdorff locally convex bornological spaces are bornological.[10]
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.[10]
• Fréchet Montel spaces have bornological strong duals.
• The strong dual of every reflexive Fréchet space is bornological.[12]
• If the strong dual of a metrizable locally convex space is separable, then it is bornological.[12]
• A vector subspace of a Hausdorff locally convex bornological space ${\displaystyle X}$  that has finite codimension in ${\displaystyle X}$  is bornological.[4][10]
• The finest locally convex topology on a vector space is bornological.[4]
Counterexamples

There exists a bornological LB-space whose strong bidual is not bornological.[13]

A closed vector subspace of a locally convex bornological space is not necessarily bornological.[4][14] There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.[4]

Bornological spaces need not be barrelled and barrelled spaces need not be bornological.[4] Because every locally convex ultrabornological space is barrelled,[4] it follows that a bornological space is not necessarily ultrabornological.

### Properties

• The strong dual space of a locally convex bornological space is complete.[4]
• Every locally convex bornological space is infrabarrelled.[4]
• Every Hausdorff sequentially complete bornological TVS is ultrabornological.[4]
• Thus every complete Hausdorff bornological space is ultrabornological.
• In particular, every Fréchet space is ultrabornological.[4]
• The finite product of locally convex ultrabornological spaces is ultrabornological.[4]
• Every Hausdorff bornological space is quasi-barrelled.[15]
• Given a bornological space ${\displaystyle X}$  with continuous dual ${\displaystyle X^{\prime },}$  the topology of ${\displaystyle X}$  coincides with the Mackey topology ${\displaystyle \tau (X,X^{\prime }).}$
• 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 ${\displaystyle X}$  be a metrizable locally convex space with continuous dual ${\displaystyle X^{\prime }.}$  Then the following are equivalent:
1. ${\displaystyle \beta (X^{\prime },X)}$  is bornological.
2. ${\displaystyle \beta (X^{\prime },X)}$  is quasi-barrelled.
3. ${\displaystyle \beta (X^{\prime },X)}$  is barrelled.
4. ${\displaystyle X}$  is a distinguished space.
• If ${\displaystyle L:X\to Y}$  is a linear map between locally convex spaces and if ${\displaystyle X}$  is bornological, then the following are equivalent:
1. ${\displaystyle L:X\to Y}$  is continuous.
2. ${\displaystyle L:X\to Y}$  is sequentially continuous.[4]
3. For every set ${\displaystyle B\subseteq X}$  that's bounded in ${\displaystyle X,}$  ${\displaystyle L(B)}$  is bounded.
4. If ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$  is a null sequence in ${\displaystyle X}$  then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}$  is a null sequence in ${\displaystyle Y.}$
5. If ${\displaystyle x_{\bullet }=(x_{i})_{i=1}^{\infty }}$  is a Mackey convergent null sequence in ${\displaystyle X}$  then ${\displaystyle L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }}$  is a bounded subset of ${\displaystyle Y.}$
• Suppose that ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex TVSs and that the space of continuous linear maps ${\displaystyle L_{b}(X;Y)}$  is endowed with the topology of uniform convergence on bounded subsets of ${\displaystyle X.}$  If ${\displaystyle X}$  is a bornological space and if ${\displaystyle Y}$  is complete then ${\displaystyle L_{b}(X;Y)}$  is a complete TVS.[4]
• In particular, the strong dual of a locally convex bornological space is complete.[4] However, it need not be bornological.
Subsets
• In a locally convex bornological space, every convex bornivorous set ${\displaystyle B}$  is a neighborhood of ${\displaystyle 0}$  (${\displaystyle B}$  is not required to be a disk).[4]
• Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.[4]
• Closed vector subspaces of bornological space need not be bornological.[4]

## Ultrabornological spaces

A disk in a topological vector space ${\displaystyle X}$  is called infrabornivorous if it absorbs all Banach disks.

If ${\displaystyle X}$  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 equivalent conditions hold:

1. Every infrabornivorous disk is a neighborhood of the origin.
2. ${\displaystyle X}$  is the inductive limit of the spaces ${\displaystyle X_{D}}$  as ${\displaystyle D}$  varies over all compact disks in ${\displaystyle X.}$
3. A seminorm on ${\displaystyle X}$  that is bounded on each Banach disk is necessarily continuous.
4. For every locally convex space ${\displaystyle Y}$  and every linear map ${\displaystyle u:X\to Y,}$  if ${\displaystyle u}$  is bounded on each Banach disk then ${\displaystyle u}$  is continuous.
5. For every Banach space ${\displaystyle Y}$  and every linear map ${\displaystyle u:X\to Y,}$  if ${\displaystyle u}$  is bounded on each Banach disk then ${\displaystyle u}$  is continuous.

### Properties

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

## References

1. ^ a b Narici & Beckenstein 2011, p. 168.
2. Narici & Beckenstein 2011, pp. 156–175.
3. ^ Wilansky 2013, p. 50.
4. Narici & Beckenstein 2011, pp. 441–457.
5. ^ Swartz 1992, pp. 15–16.
6. Narici & Beckenstein 2011, pp. 453–454.
7. ^ a b Adasch, Ernst & Keim 1978, pp. 60–61.
8. ^ Wilansky 2013, p. 48.
9. ^ a b Narici & Beckenstein 2011, p. 450.
10. Adasch, Ernst & Keim 1978, pp. 60–65.
11. ^ Narici & Beckenstein 2011, p. 453.
12. ^ a b Schaefer & Wolff 1999, p. 144.
13. ^ Khaleelulla 1982, pp. 28–63.
14. ^ Schaefer & Wolff 1999, pp. 103–110.
15. ^ Adasch, Ernst & Keim 1978, pp. 70–73.

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