Bornological space

Inmathematics,particularly infunctional analysis,abornological spaceis a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness ofsetsandlinear maps,in the same way that atopological spacepossesses the minimum amount of structure needed to address questions ofcontinuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into anylocally convexspaces is continuous if and only if it is abounded linear operator.

Bornological spaces were first studied byGeorge Mackey.The name was coined byBourbakiafterborné,the French word for "bounded".

Bornologies and bounded maps

Main article:Bornology

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

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

Elements of the collection${\mathcal {B}}$are called${\mathcal {B}}$-boundedor simplybounded setsif${\mathcal {B}}$is understood. The pair$(X,{\mathcal {B}})$is called abounded structureor abornological set.

Abaseorfundamental systemof a bornology${\mathcal {B}}$is a subset${\mathcal {B}}_{0}$of${\mathcal {B}}$such that each element of${\mathcal {B}}$is a subset of some element of${\mathcal {B}}_{0}.$Given a collection${\mathcal {S}}$of subsets of$X,$the smallest bornology containing${\mathcal {S}}$is called thebornology generated by${\mathcal {S}}.$

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

Bounded maps

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

Vector bornologies

Main article:Vector bornology

Let$X$be a vector space over afield$\mathbb {K}$where$\mathbb {K}$has a bornology${\mathcal {B}}_{\mathbb {K} }.$ A bornology${\mathcal {B}}$on$X$is called avector bornology on$X$if it is stable under vector addition, scalar multiplication, and the formation ofbalanced hulls(i.e. if the sum of two bounded sets is bounded, etc.).

If$X$is atopological vector space(TVS) and${\mathcal {B}}$is a bornology on$X,$then the following are equivalent:

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

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

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

Bornivorous subsets

A subset$A$of$X$is calledbornivorousand abornivoreif itabsorbsevery 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.

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

Every bornivorous subset of a locally convexmetrizable topological vector spaceis a neighborhood of the origin.

Mackey convergence

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

Bornology of a topological vector space

Everytopological vector space$X,$at least on a non discretevalued fieldgives a bornology on$X$by defining a subset$B\subseteq X$to bebounded(or von-Neumann bounded), if and only if for all open sets$U\subseteq X$containing zero there exists a$r>0$with$B\subseteq rU.$ If$X$is alocally convextopological vector spacethen$B\subseteq X$is bounded if and only if all continuous semi-norms on$X$are bounded on$B.$

The set of allboundedsubsets of a topological vector space$X$is calledthe bornologyorthe von Neumann bornologyof$X.$

If$X$is alocally convex topological vector space,then anabsorbingdisk$D$in$X$is bornivorous (resp. infrabornivorous) if and only if itsMinkowski functionalis locally bounded (resp. infrabounded).

Induced topology

If${\mathcal {B}}$is a convex vector bornology on a vector space$X,$then the collection${\mathcal {N}}_{\mathcal {B}}(0)$of all convexbalancedsubsets of$X$that are bornivorous forms aneighborhood basisat the origin for alocally convextopology on$X$called thetopology induced by${\mathcal {B}}$.

If$(X,\tau )$is a TVS then thebornological space associated with$X$is the vector space$X$endowed with the locally convex topology induced by the von Neumann bornology of$(X,\tau ).$

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

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

Quasi-bornological spaces

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

Atopological vector space(TVS)$(X,\tau )$with acontinuous dual$X^{\prime }$is called aquasi-bornological spaceif any of the following equivalent conditions holds:

1. Everybounded linear operatorfrom$X$into another TVS iscontinuous.
2. Every bounded linear operator from$X$into acomplete metrizable TVSis continuous.
3. Every knot in a bornivorous string is a neighborhood of the origin.

Everypseudometrizable TVSis quasi-bornological. A TVS$(X,\tau )$in which everybornivorous setis a neighborhood of the origin is a quasi-bornological space. If$X$is a quasi-bornological TVS then the finest locally convex topology on$X$that is coarser than$\tau$makes$X$into a locally convex bornological space.

Bornological space

In functional analysis, alocally convex topological vector spaceis 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 arenotquasi-bornological.

Atopological vector space(TVS)$(X,\tau )$with acontinuous dual$X^{\prime }$is called abornological spaceif it is locally convex and any of the following equivalent conditions holds:

1. Every convex, balanced, and bornivorous set in$X$is a neighborhood of zero.
2. Everybounded linear operatorfrom$X$into a locally convex TVS iscontinuous.
   • Recall that a linear map is bounded if and only if it maps any sequence converging to$0$in the domain to a bounded subset of the codomain.In particular, any linear map that is sequentially continuous at the origin is bounded.
3. Every bounded linear operator from$X$into aseminormed spaceis continuous.
4. Every bounded linear operator from$X$into aBanach spaceis continuous.

If$X$is aHausdorfflocally convex spacethen we may add to this list:

5. The locally convex topologyinduced bythe von Neumann bornology on$X$is the same as$\tau,$$X$'s given topology.
6. Every boundedseminormon$X$is continuous.
7. Any other Hausdorff locally convex topological vector space topology on$X$that has the same (von Neumann) bornology as$(X,\tau )$is necessarily coarser than$\tau.$
8. $X$is the inductive limit of normed spaces.
9. $X$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$X$).
10. $X$carries the Mackey topology$\tau (X,X^{\prime })$and all bounded linear functionals on$X$are continuous.
11. $X$has both of the following properties:
    • $X$isconvex-sequentialorC-sequential,which means that every convex sequentially open subset of$X$is open,
    • $X$issequentially bornologicalorS-bornological,which means that every convex and bornivorous subset of$X$is sequentially open.
    where a subset$A$of$X$is calledsequentially openif every sequence converging to$0$eventually belongs to$A.$

Every sequentially continuous linear operator from a locally convex bornological space into a locally convex TVS is continuous,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$F:X\to Y$from a locally convex bornological space into a locally convex space$Y$that maps null sequences in$X$tobounded subsetsof$Y$is necessarily continuous.

Sufficient conditions

Mackey–Ulam theorem—The product of a collection$X_{\bullet }=(X_{i})_{i\in I}$locally convex bornological spaces is bornological if and only if$I$doesnotadmit anUlam measure.

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

The following topological vector spaces are all bornological:

• Any locally convexpseudometrizable TVSis bornological.
  • Thus everynormed spaceandFréchet spaceis bornological.
• Any strict inductive limit of bornological spaces, in particular anystrictLF-space,is bornological.
  • This shows that there are bornological spaces that are not metrizable.
• A countable product of locally convex bornological spaces is bornological.
• Quotients of Hausdorff locally convex bornological spaces are bornological.
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.
• FréchetMontelspaces have bornologicalstrong duals.
• The strong dual of everyreflexiveFréchet spaceis bornological.
• If the strong dual of a metrizable locally convex space isseparable,then it is bornological.
• A vector subspace of a Hausdorff locally convex bornological space$X$that has finite codimension in$X$is bornological.
• Thefinest locally convex topologyon a vector space is bornological.

Counterexamples

There exists a bornologicalLB-spacewhose strong bidual isnotbornological.

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

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

Properties

• Thestrong dual spaceof a locally convex bornological space iscomplete.
• Every locally convex bornological space isinfrabarrelled.
• Every Hausdorff sequentially complete bornological TVS isultrabornological.
  • Thus everycompleteHausdorff bornological space is ultrabornological.
  • In particular, everyFréchet spaceis ultrabornological.
• The finite product of locally convex ultrabornological spaces is ultrabornological.
• Every Hausdorff bornological space isquasi-barrelled.
• Given a bornological space$X$withcontinuous dual$X^{\prime },$the topology of$X$coincides with theMackey topology$\tau (X,X^{\prime }).$
  • In particular, bornological spaces areMackey spaces.
• Everyquasi-complete(i.e. all closed and bounded subsets are complete) bornological space isbarrelled.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$X^{\prime }.$Then the following are equivalent:
  1. $\beta (X^{\prime },X)$is bornological.
  2. $\beta (X^{\prime },X)$isquasi-barrelled.
  3. $\beta (X^{\prime },X)$isbarrelled.
  4. $X$is adistinguished space.
• If$L:X\to Y$is a linear map between locally convex spaces and if$X$is bornological, then the following are equivalent:
  1. $L:X\to Y$is continuous.
  2. $L:X\to Y$is sequentially continuous.
  3. For every set$B\subseteq X$that's bounded in$X,$$L(B)$is bounded.
  4. If$x_{\bullet }=(x_{i})_{i=1}^{\infty }$is a null sequence in$X$then$L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }$is a null sequence in$Y.$
  5. If$x_{\bullet }=(x_{i})_{i=1}^{\infty }$is a Mackey convergent null sequence in$X$then$L\circ x_{\bullet }=(L(x_{i}))_{i=1}^{\infty }$is a bounded subset of$Y.$
• Suppose that$X$and$Y$arelocally convexTVSs and that the space of continuous linear maps$L_{b}(X;Y)$is endowed with thetopology of uniform convergence on bounded subsetsof$X.$If$X$is a bornological space and if$Y$iscompletethen$L_{b}(X;Y)$is a complete TVS.
  • In particular, the strong dual of a locally convex bornological space is complete.However, it need not be bornological.

Subsets

• In a locally convex bornological space, every convex bornivorous set$B$is a neighborhood of$0$($B$isnotrequired to be a disk).
• Every bornivorous subset of a locally convexmetrizable topological vector spaceis a neighborhood of the origin.
• Closed vector subspaces of bornological space need not be bornological.

Ultrabornological spaces

Main article:Ultrabornological space

A disk in a topological vector space$X$is calledinfrabornivorousif it absorbs allBanach 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 calledultrabornologicalif any of the following equivalent conditions hold:

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

• Bornology– Mathematical generalization of boundedness
• Bornivorous set– A set that can absorb any bounded subset
• Bounded set (topological vector space)– Generalization of boundedness
• Locally convex topological vector space– A vector space with a topology defined by convex open sets
• Space of linear maps
• Topological vector space– Vector space with a notion of nearness
• Vector bornology

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