Bornological space#Mackey convergence

{{Main|Bornology}}

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

1. $\backslash mathcal\{B\}$ covers $X;$ that is, $X\; =\; \backslash cup\; \backslash mathcal\{B\}$;
2. $\backslash mathcal\{B\}$ is stable under inclusions; that is, if $B\; \backslash in\; \backslash mathcal\{B\}$ and $A\; \backslash subseteq\; B,$ then $A\; \backslash in\; \backslash mathcal\{B\}$;
3. $\backslash mathcal\{B\}$ is stable under finite unions; that is, if $B\_1,\; \backslash ldots,\; B\_n\; \backslash in\; \backslash mathcal\{B\}$ then $B\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; B\_n\; \backslash in\; \backslash mathcal\{B\}$;

Elements of the collection $\backslash mathcal\{B\}$ are called {{em|$\backslash mathcal\{B\}$-bounded}} or simply {{em|bounded sets}} if $\backslash mathcal\{B\}$ is understood.{{sfn|Narici|Beckenstein|2011|p=168}}

The pair $(X,\; \backslash mathcal\{B\})$ is called a {{em|bounded structure}} or a {{em|bornological set}}.{{sfn|Narici|Beckenstein|2011|p=168}}

A {{em|base}} or {{em|fundamental system}} of a bornology $\backslash mathcal\{B\}$ is a subset $\backslash mathcal\{B\}\_0$ of $\backslash mathcal\{B\}$ such that each element of $\backslash mathcal\{B\}$ is a subset of some element of $\backslash mathcal\{B\}\_0.$ Given a collection $\backslash mathcal\{S\}$ of subsets of $X,$ the smallest bornology containing $\backslash mathcal\{S\}$ is called the {{em|bornology generated by $\backslash mathcal\{S\}.$}}{{sfn|Narici|Beckenstein|2011|pp=156–175}}

If $(X,\; \backslash mathcal\{B\})$ and $(Y,\; \backslash mathcal\{C\})$ are bornological sets then their {{em|product bornology}} on $X\; \backslash times\; Y$ is the bornology having as a base the collection of all sets of the form $B\; \backslash times\; C,$ where $B\; \backslash in\; \backslash mathcal\{B\}$ and $C\; \backslash in\; \backslash mathcal\{C\}.${{sfn|Narici|Beckenstein|2011|pp=156–175}}

A subset of $X\; \backslash times\; Y$ is bounded in the product bornology if and only if its image under the canonical projections onto $X$ and $Y$ are both bounded.

If $(X,\; \backslash mathcal\{B\})$ and $(Y,\; \backslash mathcal\{C\})$ are bornological sets then a function $f\; :\; X\; \backslash to\; Y$ is said to be a {{em|locally bounded map}} or a {{em|bounded map}} (with respect to these bornologies) if it maps $\backslash mathcal\{B\}$-bounded subsets of $X$ to $\backslash mathcal\{C\}$-bounded subsets of $Y;$ that is, if $f(\backslash mathcal\{B\})\; \backslash subseteq\; \backslash mathcal\{C\}.${{sfn|Narici|Beckenstein|2011|pp=156–175}}

If in addition $f$ is a bijection and $f^\{-1\}$ is also bounded then $f$ is called a {{em|bornological isomorphism}}.

{{Main|Vector bornology}}

Let $X$ be a vector space over a field $\backslash mathbb\{K\}$ where $\backslash mathbb\{K\}$ has a bornology $\backslash mathcal\{B\}\_\{\backslash mathbb\{K\}\}.$

A bornology $\backslash mathcal\{B\}$ on $X$ is called a {{em|vector bornology on $X$}} 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 $X$ is a topological vector space (TVS) and $\backslash mathcal\{B\}$ is a bornology on $X,$ then the following are equivalent:

1. $\backslash mathcal\{B\}$ is a vector bornology;
2. Finite sums and balanced hulls of $\backslash mathcal\{B\}$-bounded sets are $\backslash mathcal\{B\}$-bounded;{{sfn|Narici|Beckenstein|2011|pp=156–175}}
3. The scalar multiplication map $\backslash mathbb\{K\}\; \backslash times\; X\; \backslash to\; X$ defined by $(s,\; x)\; \backslash mapsto\; sx$ and the addition map $X\; \backslash times\; X\; \backslash to\; X$ defined by $(x,\; y)\; \backslash mapsto\; x\; +\; y,$ are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets).{{sfn|Narici|Beckenstein|2011|pp=156–175}}

A vector bornology $\backslash mathcal\{B\}$ is called a {{em|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 $\backslash mathcal\{B\}.$

And a vector bornology $\backslash mathcal\{B\}$ is called {{em|separated}} if the only bounded vector subspace of $X$ is the 0-dimensional trivial space $\backslash \{\; 0\; \backslash \}.$

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

A subset $A$ of $X$ is called {{em|bornivorous}} and a {{em|bornivore}} 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.{{sfn|Wilansky|2013|p=50}}

Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

A sequence $x\_\backslash bull\; =\; (x\_i)\_\{i=1\}^\backslash infty$ in a TVS $X$ is said to be {{em|Mackey convergent to $0$}} if there exists a sequence of positive real numbers $r\_\backslash bull\; =\; (r\_i)\_\{i=1\}^\backslash infty$ diverging to $\backslash infty$ such that $(r\_i\; x\_i)\_\{i=1\}^\backslash infty$ converges to $0$ in $X.${{sfn|Swartz|1992|pp=15–16}}

Every topological vector space $X,$ at least on a non discrete valued field gives a bornology on $X$ by defining a subset $B\; \backslash subseteq\; X$ to be bounded (or von-Neumann bounded), if and only if for all open sets $U\; \backslash subseteq\; X$ containing zero there exists a $r\; >\; 0$ with $B\; \backslash subseteq\; rU.$

If $X$ is a locally convex topological vector space then $B\; \backslash subseteq\; X$ 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 {{em|the bornology}} or {{em|the von Neumann bornology}} 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).{{sfn|Narici|Beckenstein|2011|pp=441–457}}

If $\backslash mathcal\{B\}$ is a convex vector bornology on a vector space $X,$ then the collection $\backslash mathcal\{N\}\_\{\backslash mathcal\{B\}\}(0)$ 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 {{em|topology induced by $\backslash mathcal\{B\}$}}.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

If $(X,\; \backslash tau)$ is a TVS then the {{em|bornological space associated with $X$}} is the vector space $X$ endowed with the locally convex topology induced by the von Neumann bornology of $(X,\; \backslash tau).${{sfn|Narici|Beckenstein|2011|pp=441–457}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=441–457}}|math_statement=

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 $L\; :\; X\; \backslash to\; Y$ is a bounded linear operator if and only if $L\; :\; X\_b\; \backslash to\; Y$ is continuous.

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

}}

Quasi-bornological spaces where introduced by S. Iyahen in 1968.{{sfn|Narici|Beckenstein|2011|pp=453–454}}

A topological vector space (TVS) $(X,\; \backslash tau)$ with a continuous dual $X^\{\backslash prime\}$ is called a {{em|quasi-bornological space}}{{sfn|Narici|Beckenstein|2011|pp=453–454}} if any of the following equivalent conditions holds:

1. Every bounded linear operator from $X$ into another TVS is continuous.{{sfn|Narici|Beckenstein|2011|pp=453–454}}
2. Every bounded linear operator from $X$ into a complete metrizable TVS is continuous.{{sfn|Narici|Beckenstein|2011|pp=453–454}}{{sfn|Adasch|Ernst|Keim|1978|pp=60–61}}
3. Every knot in a bornivorous string is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=453–454}}

Every pseudometrizable TVS is quasi-bornological. {{sfn|Narici|Beckenstein|2011|pp=453–454}}

A TVS $(X,\; \backslash tau)$ in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space.{{sfn|Wilansky|2013|p=48}}

If $X$ is a quasi-bornological TVS then the finest locally convex topology on $X$ that is coarser than $\backslash tau$ makes $X$ into a locally convex 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 {{em|not}} quasi-bornological.{{sfn|Narici|Beckenstein|2011|pp=453–454}}

A topological vector space (TVS) $(X,\; \backslash tau)$ with a continuous dual $X^\{\backslash prime\}$ is called a {{em|bornological space}} 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.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
2. Every bounded linear operator from $X$ into a locally convex TVS is continuous.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
   • 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.{{sfn|Narici|Beckenstein|2011|pp=441–457}} 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.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
4. Every bounded linear operator from $X$ into a Banach space is continuous.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

If $X$ is a Hausdorff locally convex space then we may add to this list:{{sfn|Adasch|Ernst|Keim|1978|pp=60–61}}

5. The locally convex topology induced by the von Neumann bornology on $X$ is the same as $\backslash tau,$ $X$'s given topology.
6. Every bounded seminorm on $X$ is continuous.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
7. Any other Hausdorff locally convex topological vector space topology on $X$ that has the same (von Neumann) bornology as $(X,\; \backslash tau)$ is necessarily coarser than $\backslash tau.$
8. $X$ is the inductive limit of normed spaces.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
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 $X$).{{sfn|Narici|Beckenstein|2011|pp=441–457}}
10. $X$ carries the Mackey topology $\backslash tau(X,\; X^\{\backslash prime\})$ and all bounded linear functionals on $X$ are continuous.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

11. $X$ has both of the following properties:

    • $X$ is {{em|convex-sequential}} or {{em|C-sequential}}, which means that every convex sequentially open subset of $X$ is open,
    • $X$ is {{em|sequentially bornological}} or {{em|S-bornological}}, which means that every convex and bornivorous subset of $X$ is sequentially open.

    where a subset $A$ of $X$ is called {{em|sequentially open}} 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,{{sfn|Narici|Beckenstein|2011|pp=441–457}} 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 $F\; :\; X\; \backslash to\; Y$ 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.

{{Math theorem|name=Mackey–Ulam theorem{{sfn|Narici|Beckenstein|2011|p=450}}|math_statement=

The product of a collection $X\_\backslash bull\; =\; (\; X\_i)\_\{i\; \backslash in\; I\}$ locally convex bornological spaces is bornological if and only if $I$ does {{em|not}} admit an Ulam measure.

}}

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

is bornological."{{sfn|Narici|Beckenstein|2011|p=450}}

The following topological vector spaces are all bornological:

• Any locally convex pseudometrizable TVS is bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}{{sfn|Adasch|Ernst|Keim|1978|pp=60–65}}
  • 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.{{sfn|Narici|Beckenstein|2011|p=453}}{{sfn|Adasch|Ernst|Keim|1978|pp=60–65}}
• Quotients of Hausdorff locally convex bornological spaces are bornological.{{sfn|Adasch|Ernst|Keim|1978|pp=60–65}}
• The direct sum and inductive limit of Hausdorff locally convex bornological spaces is bornological.{{sfn|Adasch|Ernst|Keim|1978|pp=60–65}}
• Fréchet Montel spaces have bornological strong duals.
• The strong dual of every reflexive Fréchet space is bornological.{{sfn|Schaefer|Wolff|1999|p=144}}
• If the strong dual of a metrizable locally convex space is separable, then it is bornological.{{sfn|Schaefer|Wolff|1999|p=144}}
• A vector subspace of a Hausdorff locally convex bornological space $X$ that has finite codimension in $X$ is bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}{{sfn|Adasch|Ernst|Keim|1978|pp=60–65}}
• The finest locally convex topology on a vector space is bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

;Counterexamples

There exists a bornological LB-space whose strong bidual is {{em|not}} bornological.{{sfn|Khaleelulla|1982|pp=28–63}}

A closed vector subspace of a locally convex bornological space is not necessarily bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}{{sfn|Schaefer|Wolff|1999|pp=103–110}}

There exists a closed vector subspace of a locally convex bornological space that is complete (and so sequentially complete) but neither barrelled nor bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

Bornological spaces need not be barrelled and barrelled spaces need not be bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}} Because every locally convex ultrabornological space is barrelled,{{sfn|Narici|Beckenstein|2011|pp=441–457}} it follows that a bornological space is not necessarily ultrabornological.

• The strong dual space of a locally convex bornological space is complete.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
• Every locally convex bornological space is infrabarrelled.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
• Every Hausdorff sequentially complete bornological TVS is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
  • Thus every complete Hausdorff bornological space is ultrabornological.
  • In particular, every Fréchet space is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

• The finite product of locally convex ultrabornological spaces is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
• Every Hausdorff bornological space is quasi-barrelled.{{sfn|Adasch|Ernst|Keim|1978|pp=70–73}}
• Given a bornological space $X$ with continuous dual $X^\{\backslash prime\},$ the topology of $X$ coincides with the Mackey topology $\backslash tau(X,\; X^\{\backslash prime\}).$
  • 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 $X^\{\backslash prime\}.$ Then the following are equivalent:

  1. $\backslash beta(X^\{\backslash prime\},\; X)$ is bornological.
  2. $\backslash beta(X^\{\backslash prime\},\; X)$ is quasi-barrelled.
  3. $\backslash beta(X^\{\backslash prime\},\; X)$ is barrelled.
  4. $X$ is a distinguished space.

• If $L\; :\; X\; \backslash to\; Y$ is a linear map between locally convex spaces and if $X$ is bornological, then the following are equivalent:

  1. $L\; :\; X\; \backslash to\; Y$ is continuous.
  2. $L\; :\; X\; \backslash to\; Y$ is sequentially continuous.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
  3. For every set $B\; \backslash subseteq\; X$ that's bounded in $X,$ $L(B)$ is bounded.
  4. If $x\_\{\backslash bull\}\; =\; (x\_i)\_\{i=1\}^\backslash infty$ is a null sequence in $X$ then $L\; \backslash circ\; x\_\backslash bull\; =\; (L(x\_i))\_\{i=1\}^\backslash infty$ is a null sequence in $Y.$
  5. If $x\_\backslash bull\; =\; (x\_i)\_\{i=1\}^\backslash infty$ is a Mackey convergent null sequence in $X$ then $L\; \backslash circ\; x\_\backslash bull\; =\; (L(x\_i))\_\{i=1\}^\backslash infty$ 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.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
  • In particular, the strong dual of a locally convex bornological space is complete.{{sfn|Narici|Beckenstein|2011|pp=441–457}} 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 {{em|not}} required to be a disk).{{sfn|Narici|Beckenstein|2011|pp=441–457}}
• Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=441–457}}
• Closed vector subspaces of bornological space need not be bornological.{{sfn|Narici|Beckenstein|2011|pp=441–457}}

{{Main|Ultrabornological space}}

A disk in a topological vector space $X$ is called {{em|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 {{em|ultrabornological}} 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 $u\; :\; X\; \backslash to\; Y,$ if $u$ is bounded on each Banach disk then $u$ is continuous.
5. For every Banach space $Y$ and every linear map $u\; :\; X\; \backslash to\; Y,$ if $u$ is bounded on each Banach disk then $u$ is continuous.

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

• {{annotated link|Bornology}}
• {{annotated link|Bornivorous set}}
• {{annotated link|Bounded set (topological vector space)}}
• {{annotated link|Locally convex topological vector space}}
• {{annotated link|Space of linear maps}}
• {{annotated link|Topological vector space}}
• {{annotated link|Vector bornology}}

{{reflist}}

• {{Adasch Topological Vector Spaces|edition=2}}
• {{Berberian Lectures in Functional Analysis and Operator Theory}}
• {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
• {{Conway A Course in Functional Analysis|edition=2}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Topological Vector Spaces}}
• {{cite book|last=Hogbe-Nlend|first=Henri|title=Bornologies and functional analysis|publisher=North-Holland Publishing Co.|location=Amsterdam|year=1977|pages=xii+144|isbn=0-7204-0712-5|mr=0500064}}
• {{Hogbe-Nlend Bornologies and Functional Analysis}}
• {{Jarchow Locally Convex Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Cite book|last1=Kriegl|first1=Andreas|last2=Michor|first2=Peter W.|title=The Convenient Setting of Global Analysis|year=1997|publisher=American Mathematical Society|series=Mathematical Surveys and Monographs|isbn=9780821807804}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{Functional analysis}}

{{Boundedness and bornology}}

{{Topological vector spaces}}

Category:Topological vector spaces