Bornological space

Template:Short description 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.Template:Fact The name was coined by BourbakiTemplate:Fact after Template:Wikt-lang, the French word for "bounded".

Bornologies and bounded maps

A Template:Em on a set $X$ is a collection $ℬ$ of subsets of $X$ that satisfy all the following conditions:

1. $ℬ$ covers $X;$ that is, $X=\cup ℬ$;
2. $ℬ$ is stable under inclusions; that is, if $B\in ℬ$ and $A\subseteq B,$ then $A\in ℬ$;
3. $ℬ$ is stable under finite unions; that is, if $B_1,\dots ,B_n\in ℬ$ then $B_1\cup \cdots \cup B_n\in ℬ$;

Elements of the collection $ℬ$ are called Template:Em or simply Template:Em if $ℬ$ is understood.Template:Sfn The pair $(X,ℬ)$ is called a Template:Em or a Template:Em.Template:Sfn

A Template:Em or Template:Em of a bornology $ℬ$ is a subset ${ℬ}_0$ of $ℬ$ such that each element of $ℬ$ is a subset of some element of ${ℬ}_0.$ Given a collection $𝒮$ of subsets of $X,$ the smallest bornology containing $𝒮$ is called the Template:EmTemplate:Sfn

If $(X,ℬ)$ and $(Y,𝒞)$ are bornological sets then their Template:Em on $X\times Y$ is the bornology having as a base the collection of all sets of the form $B\times C,$ where $B\in ℬ$ and $C\in 𝒞.$Template:Sfn 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 $(X,ℬ)$ and $(Y,𝒞)$ are bornological sets then a function $f:X\to Y$ is said to be a Template:Em or a Template:Em (with respect to these bornologies) if it maps $ℬ$-bounded subsets of $X$ to $𝒞$-bounded subsets of $Y;$ that is, if $f(ℬ)\subseteq 𝒞.$Template:Sfn If in addition $f$ is a bijection and $f^{-1}$ is also bounded then $f$ is called a Template:Em.

Vector bornologies

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

1. $ℬ$ is a vector bornology;
2. Finite sums and balanced hulls of $ℬ$-bounded sets are $ℬ$-bounded;Template:Sfn
3. The scalar multiplication map $𝕂\times X\to X$ 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).Template:Sfn

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

Usually, $𝕂$ is either the real or complex numbers, in which case a vector bornology $ℬ$ on $X$ will be called a Template:Em if $ℬ$ has a base consisting of convex sets.

Bornivorous subsets

A subset $A$ of $X$ is called Template:Em and a Template:Em 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.

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

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.Template:Sfn

Mackey convergence

A sequence $x_{\bullet }=(x_i{)}_{i=1}^{\mathrm{\infty }}$ in a TVS $X$ is said to be Template:Em if there exists a sequence of positive real numbers $r_{\bullet }=(r_i{)}_{i=1}^{\mathrm{\infty }}$ diverging to $\mathrm{\infty }$ such that $(r_i{x}_i{)}_{i=1}^{\mathrm{\infty }}$ converges to $0$ in $X.$Template:Sfn

Bornology of a topological vector space

Every topological vector space $X,$ at least on a non discrete valued field gives a bornology on $X$ by defining a subset $B\subseteq X$ to be bounded (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 a locally convex topological vector space then $B\subseteq X$ is bounded if and only if all continuous semi-norms on $X$ are bounded on $B.$

The set of all bounded subsets of a topological vector space $X$ is called Template:Em or Template:Em of $X.$

If $X$ is a locally convex topological vector space, then an absorbing disk $D$ in $X$ is bornivorous (resp. infrabornivorous) if and only if its Minkowski functional is locally bounded (resp. infrabounded).Template:Sfn

Induced topology

If $ℬ$ is a convex vector bornology on a vector space $X,$ then the collection ${𝒩}_{ℬ}(0)$ of all convex balanced subsets of $X$ that are bornivorous forms a neighborhood basis at the origin for a locally convex topology on $X$ called the Template:Em.Template:Sfn

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

Quasi-bornological spaces

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

A topological vector space (TVS) $(X,\tau )$ with a continuous dual $X^{\prime }$ is called a Template:EmTemplate:Sfn if any of the following equivalent conditions holds:

1. Every bounded linear operator from $X$ into another TVS is continuous.Template:Sfn
2. Every bounded linear operator from $X$ into a complete metrizable TVS is continuous.Template:SfnTemplate:Sfn
3. Every knot in a bornivorous string is a neighborhood of the origin.Template:Sfn

Every pseudometrizable TVS is quasi-bornological. Template:Sfn A TVS $(X,\tau )$ in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.Template:Sfn 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, 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 Template:Em quasi-bornological.Template:Sfn

A topological vector space (TVS) $(X,\tau )$ with a continuous dual $X^{\prime }$ is called a Template:Em if 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.Template:Sfn
2. Every bounded linear operator from $X$ into a locally convex TVS is continuous.Template:Sfn
   • 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.Template:Sfn In particular, any linear map that is sequentially continuous at the origin is bounded.
3. Every bounded linear operator from $X$ into a seminormed space is continuous.Template:Sfn
4. Every bounded linear operator from $X$ into a Banach space is continuous.Template:Sfn

If $X$ is a Hausdorff locally convex space then we may add to this list:Template:Sfn

5. The locally convex topology induced by the von Neumann bornology on $X$ is the same as $\tau ,$ $X$'s given topology.
6. Every bounded seminorm on $X$ is continuous.Template:Sfn
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.Template:Sfn
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 $X$).Template:Sfn
10. $X$ carries the Mackey topology $\tau (X,X^{\prime })$ and all bounded linear functionals on $X$ are continuous.Template:Sfn
11. $X$ has both of the following properties:
    • $X$ is Template:Em or Template:Em, which means that every convex sequentially open subset of $X$ is open,
    • $X$ is Template:Em or Template:Em, which means that every convex and bornivorous subset of $X$ is sequentially open.
    where a subset $A$ of $X$ is called Template:Em if 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,Template:Sfn 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 $X$ to bounded subsets of $Y$ is necessarily continuous.

Sufficient conditions

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

The following topological vector spaces are all bornological:

• Any locally convex pseudometrizable TVS is bornological.Template:SfnTemplate:Sfn
  • Thus every normed space and Fréchet space is bornological.
• 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.Template:SfnTemplate:Sfn
• Quotients of Hausdorff locally convex bornological spaces are bornological.Template:Sfn
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.Template:Sfn
• Fréchet Montel spaces have bornological strong duals.
• The strong dual of every reflexive Fréchet space is bornological.Template:Sfn
• If the strong dual of a metrizable locally convex space is separable, then it is bornological.Template:Sfn
• A vector subspace of a Hausdorff locally convex bornological space $X$ that has finite codimension in $X$ is bornological.Template:SfnTemplate:Sfn
• The finest locally convex topology on a vector space is bornological.Template:Sfn

Counterexamples

There exists a bornological LB-space whose strong bidual is Template:Em bornological.Template:Sfn

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

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

Properties

• The strong dual space of a locally convex bornological space is complete.Template:Sfn
• Every locally convex bornological space is infrabarrelled.Template:Sfn
• Every Hausdorff sequentially complete bornological TVS is ultrabornological.Template:Sfn
  • Thus every complete Hausdorff bornological space is ultrabornological.
  • In particular, every Fréchet space is ultrabornological.Template:Sfn
• The finite product of locally convex ultrabornological spaces is ultrabornological.Template:Sfn
• Every Hausdorff bornological space is quasi-barrelled.Template:Sfn
• Given a bornological space $X$ with continuous dual $X^{\prime },$ the topology of $X$ coincides with the Mackey topology $\tau (X,X^{\prime }).$
  • 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 $X^{\prime }.$ Then the following are equivalent:
  1. $\beta (X^{\prime },X)$ is bornological.
  2. $\beta (X^{\prime },X)$ is quasi-barrelled.
  3. $\beta (X^{\prime },X)$ is barrelled.
  4. $X$ is a distinguished 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.Template:Sfn
  3. For every set $B\subseteq X$ that's bounded in $X,$ $L(B)$ is bounded.
  4. If $x_{\bullet }=(x_i{)}_{i=1}^{\mathrm{\infty }}$ is a null sequence in $X$ then $L\circ x_{\bullet }=(L(x_i){)}_{i=1}^{\mathrm{\infty }}$ is a null sequence in $Y.$
  5. If $x_{\bullet }=(x_i{)}_{i=1}^{\mathrm{\infty }}$ is a Mackey convergent null sequence in $X$ then $L\circ x_{\bullet }=(L(x_i){)}_{i=1}^{\mathrm{\infty }}$ is a bounded subset of $Y.$
• Suppose that $X$ and $Y$ are locally convex TVSs and that the space of continuous linear maps $L_b(X;Y)$ is endowed with the topology of uniform convergence on bounded subsets of $X.$ If $X$ is a bornological space and if $Y$ is complete then $L_b(X;Y)$ is a complete TVS.Template:Sfn
  • In particular, the strong dual of a locally convex bornological space is complete.Template:Sfn However, it need not be bornological.

Subsets

• In a locally convex bornological space, every convex bornivorous set $B$ is a neighborhood of $0$ ($B$ is Template:Em required to be a disk).Template:Sfn
• Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.Template:Sfn
• Closed vector subspaces of bornological space need not be bornological.Template:Sfn

Ultrabornological spaces

A disk in a topological vector space $X$ is called Template:Em 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 Template:Em if 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. A seminorm on $X$ 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

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Bibliography

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