|
Throughout we assume the following:
<ol>
<li>{{mvar|<math>T}}</math> is any non-empty set and <math>\mathcal{{G}</math|𝒢}}> is a non-empty collection of subsets of {{mvar|<math>T}}</math> [[Directed set|directed]] by subset inclusion (i.e. for any {{<math|''>G'', ''H'' ∈\in 𝒢}\mathcal{G}</math> there exists some {{<math|''>K'' ∈\in 𝒢}\mathcal{G}</math> such that {{math|''G'' ∪ ''H'' ⊆ ''K''}}).</li>
<li>{{mvar|<math>Y}}</math> is a [[topological vector space]] (not necessarily Hausdorff or locally convex) and {{math|𝒩}} is a basis of neighborhoods of 0 in {{mvar|<math>Y}}.</math></li>
<li>{{math|''Y''<supmath>''Y^T''</supmath>}} denotes the set of all {{mvar|<math>Y}}</math>-valued functions with ___domain {{mvar|<math>T}}.</math></li> <li>{{mvar|<math>F}}</math> is a vector subspace of {{math|''Y''<supmath>''Y^T''</supmath>}} (not necessarily consisting of linear maps).</li>
</ol>
:'''Definition and notation''': For any subsets {{mvar|<math>G}}</math> of {{mvar|<math>X}}</math> and {{mvar|<math>N}}</math> of {{mvar|<math>Y}},</math> let
:"{{math|1=𝒰(''G'', ''N'') := { ''f'' ∈ ''F'' : ''f'' (''G'') ⊆ ''N''}}}.
=== Basic neighborhoods at the origin ===
Henceforth assume that {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{<math|''>N'' ∈\in 𝒩}\mathcal{N}.</math>
;Properties
<ul>
<li><math>\mathcal{{math|𝒰U}(''G'', ''N'')}}</math> is an [[Absorbing set|absorbing]] subset of {{mvar|<math>F}}</math> if and only if for all {{<math|''>f'' ∈\in ''F''}},</math> {{mvar|<math>N}}</math> absorbs {{math|''f'' (''G'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>If {{mvar|<math>N}}</math> is [[Balanced set|balanced]] then so is <math>\mathcal{{math|𝒰U}(''G'', ''N'')}}.</math>{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>If {{mvar|<math>N}}</math> is [[Convex set|convex]] then so is <math>\mathcal{{math|𝒰U}(''G'', ''N'')}}.</math></li>
</ul>
<ul>
<li>For any scalar {{mvar|s}}, {{math|1=''s''𝒰(''G'', ''N'') = 𝒰(''G'', ''sN'')}}; so in particular, {{math|1=-𝒰(''G'', ''N'') = 𝒰(''G'', -''N'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>{{math|𝒰(''G'' ∪ ''H'', ''M'' ∩ ''N'') ⊆ 𝒰(''G'', ''M'') ∩ 𝒰(''H'', ''N'')}} for any subsets {{mvar|<math>G}}</math> and {{mvar|<math>H}}</math> of {{mvar|<math>X}}</math> and non-empty subsets {{mvar|M}} and {{mvar|<math>N}}</math> of {{mvar|<math>Y}}.</math>{{sfn | Jarchow | 1981 | pp=43-55}} Thus:
<ul>
<li>If {{math|''M'' ⊆ ''N''}} then {{math|𝒰(''G'', ''M'') ⊆ 𝒰(''G'', ''N'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>If {{math|''G'' ⊆ ''H''}} then {{math|𝒰(''H'', ''N'') ⊆ 𝒰(''G'', ''N'')}}.</li>
<li>For any {{math|''M'', ''N'' ∈ 𝒩}} and subsets {{math|''G'', ''H'', ''K''}} of {{mvar|<math>T}},</math> if {{math|''G'' ∪ ''H'' ⊆ ''K''}} then {{math|𝒰(''K'', ''M'' ∩ ''N'') ⊆ 𝒰(''G'', ''M'') ∩ 𝒰(''H'', ''N'')}}.</li>
</ul>
</li>
<li>{{math|1=𝒰(∅, ''N'') = ''F''}}.</li>
<li>{{math|𝒰(''G'', ''N'') - 𝒰(''G'', ''N'') ⊆ 𝒰(''G'', ''N'' - ''N'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}</li>
<li>{{math|𝒰(''G'', ''M'') + 𝒰(''G'', ''N'') ⊆ 𝒰(''G'', ''M'' + ''N'')}}.{{sfn | Jarchow | 1981 | pp=43-55}}</li>
<li>For any family <math>\mathcal{{S}</math|𝒮}}> of subsets of {{mvar|<math>T}},</math> {{math|1=𝒰({{underset|S ∈ 𝒮|{{big|∪}}}} ''S'', ''N'') = {{underset|S ∈ 𝒮|{{big|∩}}}} 𝒰(''S'', ''N'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}</li>
<li>For any family <math>\mathcal{{M}</math|ℳ}}> of neighborhoods of 0 in {{mvar|<math>Y}},</math> {{math|1=𝒰(''G'', {{underset|M ∈ ℳ|{{big|∩}}}} ''M'') = {{underset|M ∈ ℳ|{{big|∩}}}} 𝒰(''G'', ''M'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}</li>
</ul>
=== <math>\mathcal{{G}</math|𝒢}}>-topology ===
Then the set {{math|{𝒰(''G'', ''N'') : ''G'' ∈ 𝒢, ''N'' ∈ 𝒩}}} forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set <math>\mathcal{{math|𝒰U}(''G'', ''N'')}}</math> is a neighborhood of the origin for this topology, but it is not necessarily an ''open'' neighborhood of the origin.</ref>
at the origin for a unique translation-invariant topology on {{mvar|<math>F}},</math> where this topology is ''not'' necessarily a vector topology (i.e. it might not make {{mvar|<math>F}}</math> into a TVS).
This topology does not depend on the neighborhood basis {{math|𝒩}} that was chosen and it is known as the '''topology of uniform convergence on the sets in <math>\mathcal{{G}</math|𝒢}}>''' or as the '''<math>\mathcal{{G}</math|𝒢}}>-topology'''.{{sfn | Schaefer|Wolff| 1999 | pp=79-88}}
However, this name is frequently changed according to the types of sets that make up {{<math|𝒢}>\mathcal{G}</math> (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details<ref>In practice, {{<math|𝒢}>\mathcal{G}</math> usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, {{<math|𝒢}>\mathcal{G}</math> is the collection of compact subsets of {{mvar|<math>T}}</math> (and {{mvar|<math>T}}</math> is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of {{mvar|<math>T}}.</math></ref>).
A subset {{math|𝒢<submath>1\mathcal{G}_1</submath>}} of {{<math|𝒢}>\mathcal{G}</math> is said to be '''fundamental with respect to {{<math|𝒢}>\mathcal{G}</math>''' if each {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> is a subset of some element in {{math|𝒢<submath>1\mathcal{G}_1.</submath>}}.
In this case, the collection <math>\mathcal{{G}</math|𝒢}}> can be replaced by {{<math|𝒢<sub>1\mathcal{G}_1</submath>}} without changing the topology on {{mvar|<math>F}}.</math>{{sfn | Schaefer|Wolff| 1999 | pp=79-88}}
One may also replace <math>\mathcal{{G}</math|𝒢}}> with the collection of all subsets of all finite unions of elements of <math>\mathcal{{G}</math|𝒢}}> without changing the resulting <math>\mathcal{{G}</math|𝒢}}>-topology on {{mvar|<math>F}}.</math>{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}
:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Call a subset {{mvar|<math>B}}</math> of {{mvar|<math>T}}</math> '''{{mvar|<math>F}}</math>-bounded''' if {{<math|''>f'' (''B'')}}</math> is a bounded subset of {{mvar|<math>Y}}</math> for every {{<math|''>f'' ∈\in ''F''}}.</math>
{{Math theorem|name=Theorem{{sfn | Schaefer|Wolff| 1999 | pp=79-88}}{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=
The {{<math|𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> is compatible with the vector space structure of {{mvar|<math>F}}</math> if and only if every {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> is {{mvar|<math>F}}</math>-bounded;
that is, if and only if for every {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and every {{<math|''>f'' ∈\in ''F''}},</math> {{math|''f'' (''G'')}} is [[Bounded set (topological vector space)|bounded]] in {{mvar|<math>Y}}.</math>
}}
==== Nets and uniform convergence ====
:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Let {{<math|''>f'' ∈\in ''F''}}</math> and let {{math|1=''f''<sub>•</sub> = (''f''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} be a [[Net (mathematics)|net]] in {{mvar|<math>F}}.</math> Then for any subset {{mvar|<math>G}}</math> of {{mvar|<math>T}},</math> say that {{math|''f''<sub>•</sub>}} '''converges uniformly to {{mvar|f}} on {{mvar|<math>G}}</math>''' if for every {{<math|''>N'' ∈\in 𝒩}\mathcal{N}</math> there exists some {{math|''i''<sub>0</sub> ∈ ''I''}} such that for every {{math|''i'' ∈ ''I''}} satisfying {{math|''i'' ≥ ''i''<sub>0</sub>}}, {{math|''f''<sub>''i''</sub> - ''f'' ∈ 𝒰(''G'', ''N'')}} (or equivalently, {{math|''f''<sub>''i''</sub>(''g'') - ''f'' (''g'') ∈ ''N''}} for every {{math|''g'' ∈ ''G''}}).
{{Math theorem|name=Theorem{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=
If {{<math|''>f'' ∈\in ''F''}}</math> and if {{math|1=''f''<sub>•</sub> = (''f''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} is a net in {{mvar|<math>F}},</math> then {{math|''f''<sub>•</sub> → ''f''}} in the {{<math|𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> if and only if for every {{<math|''>G'' ∈\in 𝒢}\mathcal{G},</math> {{math|''f''<sub>•</sub>}} converges uniformly to {{mvar|f}} on {{mvar|<math>G}}.</math>
}}
;Local convexity
If {{mvar|<math>Y}}</math> is [[locally convex]] then so is the {{<math|𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> and if {{math|(''p''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} is a family of continuous seminorms generating this topology on {{mvar|<math>Y}}</math> then the {{<math|𝒢}>\mathcal{G}</math>-topology is induced by the following family of seminorms:
:{{math|''p''<sub>''G'',''i''</sub>(''f'') {{=}}}} {{underset|{{math|''x'' ∈ ''G''}}|sup}} {{math|''p''<sub>''i''</sub>(''f''(''x''))}},
as {{mvar|<math>G}}</math> varies over <math>\mathcal{{G}</math|𝒢}}> and {{mvar|i}} varies over {{mvar|I}}.{{sfn | Schaefer|Wolff| 1999 | p=81}}
;Hausdorffness
If {{mvar|<math>Y}}</math> is [[Hausdorff space|Hausdorff]] and {{math|1=''T'' = {{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} then the <math>\mathcal{{G}</math|𝒢}}>-topology on {{mvar|<math>F}}</math> is Hausdorff.{{sfn | Jarchow | 1981 | pp=43-55}}
Suppose that {{mvar|<math>T}}</math> is a topological space.
If {{mvar|<math>Y}}</math> is [[Hausdorff space|Hausdorff]] and {{mvar|<math>F}}</math> is the vector subspace of {{<math|''>Y''<sup>''^T''</supmath>}} consisting of all continuous maps that are bounded on every {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and if {{math|{{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|<math>T}}</math> then the <math>\mathcal{{math|𝒢}G}</math>-topology on {{mvar|<math>F}}</math> is Hausdorff.
;Boundedness
A subset {{mvar|<math>H}}</math> of {{mvar|<math>F}}</math> is [[Bounded set (topological vector space)|bounded]] in the <math>\mathcal{{G}</math|𝒢}}>-topology if and only if for every {{<math|''>G'' ∈\in 𝒢}\mathcal{G},</math> {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|<math>Y}}.</math>{{sfn | Schaefer|Wolff| 1999 | p=81}}
=== Examples of 𝒢-topologies ===
;Pointwise convergence
If we let <math>\mathcal{{G}</math|𝒢}}> be the set of all finite subsets of {{mvar|<math>T}}</math> then the <math>\mathcal{{G}</math|𝒢}}>-topology on {{mvar|<math>F}}</math> is called the '''topology of pointwise convergence'''.
The topology of pointwise convergence on {{mvar|<math>F}}</math> is identical to the subspace topology that {{mvar|<math>F}}</math> inherits from {{math|''Y''<supmath>''Y^T''</supmath>}} when {{math|''Y''<supmath>''Y^T''</supmath>}} is endowed with the usual [[product topology]].
If {{mvar|<math>X}}</math> is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and {{math|C(''X'')}} is the space of all real (or complex) valued continuous functions on {{mvar|<math>X}},</math> the topology of pointwise convergence on {{math|C(''X'')}} is [[Metrizable TVS|metrizable]] if and only if {{mvar|<math>X}}</math> is countable.{{sfn | Jarchow | 1981 | pp=43-55}}
== 𝒢-topologies on spaces of continuous linear maps ==
Throughout this section we will assume that {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are [[topological vector space]]s.
<math>\mathcal{{G}</math|𝒢}}> will be a non-empty collection of subsets of {{mvar|<math>X}}</math> [[Directed set|directed]] by inclusion.
:'''Notation''': {{<math|>L(''X''; ''Y'')}}</math> will denote the vector space of all continuous linear maps from {{mvar|<math>X}}</math> into {{mvar|<math>Y}}.</math> If {{<math|>L(''X''; ''Y'')}}</math> is given the <math>\mathcal{{G}</math|𝒢}}>-topology inherited from {{math|''Y''<sup>''X''</sup>}} then this space with this topology is denoted by {{math|L<sub>𝒢</sub>(''X'', ''Y'')}}.
:'''Notation''': The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space {{mvar|<math>X}}</math> over the field <math>\mathbb{F}</math> (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space {{<math|>L(''X''; <math>\mathbb{F})</math>)}} and is denoted by {{math|''X''{{big|{{'}}}}}}.
The <math>\mathcal{{G}</math|𝒢}}>-topology on {{<math|>L(''X''; ''Y'')}}</math> is compatible with the vector space structure of {{<math|>L(''X''; ''Y'')}}</math> if and only if for all {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and all {{math|''f'' ∈ L(''X''; ''Y'')}} the set {{math|''f''(''G'')}} is bounded in {{mvar|<math>Y}},</math> which we will assume to be the case for the rest of the article.
Note in particular that this is the case if <math>\mathcal{{G}</math|𝒢}}> consists of [[Bounded set (topological vector space)|(von-Neumann) bounded]] subsets of {{mvar|<math>X}}.</math>
=== Assumptions on 𝒢 ===
;Assumptions that guarantee a vector topology
:'''Assumption''' (<math>\mathcal{{G}</math|𝒢}}> is directed): <math>\mathcal{{G}</math|𝒢}}> will be a non-empty collection of subsets of {{mvar|<math>X}}</math> [[Directed set|directed]] by (subset) inclusion. That is, for any {{<math|''>G'', ''H'' ∈\in 𝒢}\mathcal{G},</math> there exists {{<math|''>K'' ∈\in 𝒢}\mathcal{G}</math> such that {{math|''G'' ∪ ''H'' ⊆ ''K''}}.
The above assumption guarantees that the collection of sets <math>\mathcal{{math|𝒰U}(''G'', ''N'')}}</math> forms a [[filter base]].
The next assumption will guarantee that the sets <math>\mathcal{{math|𝒰U}(''G'', ''N'')}}</math> are [[Balanced set|balanced]].
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdonsome.
:'''Assumption''' ({{<math|''>N'' ∈\in 𝒩}\mathcal{N}</math> are balanced): {{math|𝒩}} is a neighborhoods basis of 0 in {{mvar|<math>Y}}</math> that consists entirely of [[Balanced set|balanced]] sets.
The following assumption is very commonly made because it will guarantee that each set {{<math|𝒰>\mathcal{U}(''G'', ''N'')}}</math> is absorbing in {{<math|>L(''X''; ''Y'')}}.</math>
:'''Assumption''' ({{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> are bounded): <math>\mathcal{{G}</math|𝒢}}> is assumed to consist entirely of bounded subsets of {{mvar|<math>X}}.</math>
;Other possible assumptions
The next theorem gives ways in which <math>\mathcal{{G}</math|𝒢}}> can be modified without changing the resulting <math>\mathcal{{G}</math|𝒢}}>-topology on {{mvar|<math>Y}}.</math>
{{Math theorem|name=Theorem{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}|math_statement=
Let <math>\mathcal{{G}</math|𝒢}}> be a non-empty collection of bounded subsets of {{mvar|<math>X}}.</math> Then the <math>\mathcal{{G}</math|𝒢}}>-topology on {{<math|>L(''X''; ''Y'')}}</math> is not altered if <math>\mathcal{{G}</math|𝒢}}> is replaced by any of the following collections of (also bounded) subsets of {{mvar|<math>X}}</math>:
<ol>
<li>all subsets of all finite unions of sets in <math>\mathcal{{G}</math|𝒢}}>;</li>
<li>all scalar multiples of all sets in <math>\mathcal{{G}</math|𝒢}}>;</li>
<li>all finite [[Minkowski sum]]s of sets in <math>\mathcal{{G}</math|𝒢}}>;</li>
<li>the [[Balanced set|balanced hull]] of every set in <math>\mathcal{{G}</math|𝒢}}>;</li>
<li>the closure of every set in <math>\mathcal{{G}</math|𝒢}}>;</li>
</ol>
and if {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex, then we may add to this list:
<ol start=6>
<li>the closed [[Absolutely convex|convex balanced hull]] of every set in <math>\mathcal{{math|𝒢}G}.</math></li>
</ol>
}}
;Common assumptions
Some authors (e.g. Narici) require that <math>\mathcal{{G}</math|𝒢}}> satisfy the following condition, which implies, in particular, that <math>\mathcal{{G}</math|𝒢}}> is [[Directed set|directed]] by subset inclusion:
:<math>\mathcal{{G}</math|𝒢}}> is assumed to be closed with respect to the formation of subsets of finite unions of sets in <math>\mathcal{{G}</math|𝒢}}> (i.e. every subset of every finite union of sets in <math>\mathcal{{G}</math|𝒢}}> belongs to <math>\mathcal{{G}</math|𝒢}}>).
Some authors (e.g. Trèves) require that <math>\mathcal{{G}</math|𝒢}}> be directed under subset inclusion and that it satisfy the following condition:
:If {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{mvar|s}} is a scalar then there exists a {{<math|''>H'' ∈\in 𝒢}\mathcal{G}</math> such that {{math|''sG'' ⊆ ''H''}}.
If <math>\mathcal{{G}</math|𝒢}}> is a [[bornology]] on {{mvar|<math>X}},</math> which is often the case, then these axioms are satisfied.
If <math>\mathcal{{G}</math|𝒢}}> is a [[saturated family]] of [[Bounded set (topological vector space)|bounded]] subsets of {{mvar|<math>X}}</math> then these axioms are also satisfied.
=== Properties ===
;Hausdorffness
:'''Definition''':{{sfn | Schaefer|Wolff| 1999 | p=80}} If {{mvar|<math>T}}</math> is a TVS then we say that <math>\mathcal{{G}</math|𝒢}}> is '''total in {{mvar|<math>T}}</math>''' if the [[linear span]] of {{math|{{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|<math>T}}.</math>
If {{mvar|<math>F}}</math> is the vector subspace of {{<math|''>Y''<sup>''^T''</supmath>}} consisting of all continuous linear maps that are bounded on every {{<math|''>G'' ∈\in 𝒢}\mathcal{G},</math> then the {{<math|𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> is Hausdorff if {{mvar|<math>Y}}</math> is Hausdorff and {{<math|𝒢}>\mathcal{G}</math> is total in {{mvar|<math>T}}.</math>{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}
;Completeness
For the following theorems, suppose that {{mvar|<math>X}}</math> is a topological vector space and {{mvar|<math>Y}}</math> is a [[locally convex]] Hausdorff spaces and <math>\mathcal{{G}</math|𝒢}}> is a collection of bounded subsets of {{mvar|<math>X}}</math> that covers {{mvar|<math>X}},</math> is directed by subset inclusion, and satisfies the following condition: if {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{mvar|s}} is a scalar then there exists a {{<math|''>H'' ∈\in 𝒢}\mathcal{G}</math> such that {{math|''sG'' ⊆ ''H''}}.
<ul>
<li>{{math|L<sub>𝒢</sub>(''X''; ''Y'')}} is complete if
{{ordered list|
| {{mvar|<math>X}}</math> is locally convex and Hausdorff,
| {{mvar|<math>Y}}</math> is complete, and
| whenever {{math|''u'' : ''X'' → ''Y''}} is a linear map then {{mvar|u}} restricted to every set {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> is continuous implies that {{mvar|u}} is continuous,
}}</li>
<li>If {{mvar|<math>X}}</math> is a Mackey space then {{math|L<sub>𝒢</sub>(''X''; ''Y'')}}is complete if and only if both <math>X^{\prime}_{\mathcal{G}}</math> and {{mvar|<math>Y}}</math> are complete.</li>
<li>If {{mvar|<math>X}}</math> is [[Barrelled space|barrelled]] then {{math|L<sub>𝒢</sub>(''X''; ''Y'')}} is Hausdorff and [[quasi-complete]].</li>
<li>Let {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> be TVSs with {{mvar|<math>Y}}</math> [[quasi-complete]] and assume that (1) {{mvar|<math>X}}</math> is [[barreled space|barreled]], or else (2) {{mvar|<math>X}}</math> is a [[Baire space]] and {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex. If {{<math|𝒢}>\mathcal{G}</math> covers {{mvar|<math>X}}</math> then every closed equicontinuous subset of {{<math|>L(''X''; ''Y'')}}</math> is complete in {{math|L<sub>𝒢</sub>(''X''; ''Y'')}} and {{math|L<sub>𝒢</sub>(''X''; ''Y'')}} is quasi-complete.{{sfn | Schaefer|Wolff| 1999 | p=83}}</li>
<li>Let {{mvar|<math>X}}</math> be a [[bornological space]], {{mvar|<math>Y}}</math> a locally convex space, and {{<math|𝒢}>\mathcal{G}</math> a family of bounded subsets of {{mvar|<math>X}}</math> such that the range of every null sequence in {{mvar|<math>X}}</math> is contained in some {{<math|''>G'' ∈\in 𝒢}\mathcal{G}.</math> If {{mvar|<math>Y}}</math> is [[quasi-complete]] (resp. complete) then so is {{math|L<sub>𝒢</sub>(''X''; ''Y'')}}.{{sfn | Schaefer|Wolff| 1999 | p=117}}</li>
</ul>
;Boundedness
Let {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> be topological vector spaces and {{mvar|<math>H}}</math> be a subset of {{<math|>L(''X''; ''Y'')}}.</math>
Then the following are equivalent:{{sfn | Schaefer|Wolff| 1999 | p=81}}
<ol>
<li>{{mvar|<math>H}}</math> is [[Bounded set (topological vector space)|bounded]] in {{math|L<sub>𝒢</sub>(''X''; ''Y'')}};</li>
<li>For every {{<math|''>G'' ∈\in 𝒢}\mathcal{G},</math> {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|<math>Y}}</math>;{{sfn | Schaefer|Wolff| 1999 | p=81}}</li>
<li>For every neighborhood {{mvar|V}} of 0 in {{mvar|<math>Y}}</math> the set {{math|{{underset|''h'' ∈ ''H''|{{big|∩}}}} ''h''<sup>−1</sup>(''V'')}} [[Absorbing set|absorbs]] every {{<math|''>G'' ∈\in 𝒢}\mathcal{G}.</math></li>
</ol>
Furthermore,
<ul>
<li>If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex Hausdorff space and if {{mvar|<math>H}}</math> is bounded in {{math|L<sub>𝜎</sub>(''X''; ''Y'')}} (i.e. pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of {{mvar|<math>X}}.</math>{{sfn | Schaefer|Wolff| 1999 | p=82}}</li>
<li>If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex Hausdorff spaces and if {{mvar|<math>X}}</math> is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of {{<math|>L(''X''; ''Y'')}}</math> are identical for all <math>\mathcal{{G}</math|𝒢}}>-topologies where <math>\mathcal{{G}</math|𝒢}}> is any family of bounded subsets of {{mvar|<math>X}}</math> covering {{mvar|<math>X}}.</math>{{sfn | Schaefer|Wolff| 1999 | p=82}}</li>
<li>If <math>\mathcal{{G}</math|𝒢}}> is any collection of bounded subsets of {{mvar|<math>X}}</math> whose union is total in {{mvar|<math>X}}</math> then every equicontinuous subset of {{<math|>L(''X''; ''Y'')}}</math> is bounded in the <math>\mathcal{{G}</math|𝒢}}>-topology.{{sfn | Schaefer|Wolff| 1999 | p=83}}</li>
</ul>
! Alternative name
|-
| finite subsets of {{mvar|<math>X}}</math>
| {{math|L<sub>σ</sub>(''X''; ''Y'')}}
| pointwise/simple convergence
| topology of simple convergence
|-
| precompact subsets of {{mvar|<math>X}}</math>
|
| precompact convergence
|
|-
| compact convex subsets of {{mvar|<math>X}}</math>
| {{math|L<sub>γ</sub>(''X''; ''Y'')}}
| compact convex convergence
|
|-
| compact subsets of {{mvar|<math>X}}</math>
| {{math|L<sub>c</sub>(''X''; ''Y'')}}
| compact convergence
|
|-
| bounded subsets of {{mvar|<math>X}}</math>
| {{math|L<sub>b</sub>(''X''; ''Y'')}}
| bounded convergence
==== The topology of pointwise convergence {{math|L<sub>σ</sub>(''X''; ''Y'')}} ====
By letting <math>\mathcal{{G}</math|𝒢}}> be the set of all finite subsets of {{mvar|<math>X}},</math> {{<math|>L(''X''; ''Y'')}}</math> will have the '''weak topology on {{<math|>L(''X''; ''Y'')}}</math>''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and {{<math|>L(''X''; ''Y'')}}</math> with this topology is denoted by {{math|L<sub>𝜎</sub>(''X''; ''Y'')}}.
Unfortunately, this topology is also sometimes called '''the strong operator topology''', which may lead to ambiguity;{{sfn | Narici | Beckenstein | 2011 | pp=371-423}} for this reason, this article will avoid referring to this topology by this name.
:'''Definition''': A subset of {{<math|>L(''X''; ''Y'')}}</math> is called '''simply bounded''' or '''weakly bounded''' if it is bounded in {{math|L<sub>𝜎</sub>(''X''; ''Y'')}}.
The weak-topology on {{<math|>L(''X''; ''Y'')}}</math> has the following properties:
<ul>
<li>If {{mvar|<math>X}}</math> is [[Separable space|separable]] (i.e. has a countable dense subset) and if {{mvar|<math>Y}}</math> is a metrizable topological vector space then every equicontinuous subset {{mvar|<math>H}}</math> of {{math|L<sub>𝜎</sub>(''X''; ''Y'')}} is metrizable; if in addition {{mvar|<math>Y}}</math> is separable then so is {{mvar|<math>H}}.</math>{{sfn | Schaefer|Wolff| 1999 | p=87}}
* So in particular, on every equicontinuous subset of {{<math|>L(''X''; ''Y'')}},</math> the topology of pointwise convergence is metrizable.</li>
<li>Let {{math|''Y''<sup>''X''</sup>}} denote the space of all functions from {{mvar|<math>X}}</math> into {{mvar|<math>Y}}.</math> If {{<math|>L(''X''; ''Y'')}}</math> is given the topology of pointwise convergence then space of all linear maps (continuous or not) {{mvar|<math>X}}</math> into {{mvar|<math>Y}}</math> is closed in {{math|''Y''<sup>''X''</sup>}}.
* In addition, {{<math|>L(''X''; ''Y'')}}</math> is dense in the space of all linear maps (continuous or not) {{mvar|<math>X}}</math> into {{mvar|<math>Y}}.</math></li>
<li>Suppose {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex. Any simply bounded subset of {{<math|>L(''X''; ''Y'')}}</math> is bounded when {{<math|>L(''X''; ''Y'')}}</math> has the topology of uniform convergence on convex, [[balanced set|balanced]], bounded, complete subsets of {{mvar|<math>X}}.</math> If in addition {{mvar|<math>X}}</math> is [[quasi-complete]] then the families of bounded subsets of {{<math|>L(''X''; ''Y'')}}</math> are identical for all <math>\mathcal{{G}</math|𝒢}}>-topologies on {{<math|>L(''X''; ''Y'')}}</math> such that <math>\mathcal{{G}</math|𝒢}}> is a family of bounded sets covering {{mvar|<math>X}}.</math>{{sfn | Schaefer|Wolff| 1999 | p=82}}</li>
</ul>
;Equicontinuous subsets
<ul>
<li>The weak-closure of an equicontinuous subset of {{<math|>L(''X''; ''Y'')}}</math> is equicontinuous.</li>
<li>If {{mvar|<math>Y}}</math> is locally convex, then the convex balanced hull of an equicontinuous subset of {{<math|>L(''X''; ''Y'')}}</math> is equicontinuous.</li>
<li>Let {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> be TVSs and assume that (1) {{mvar|<math>X}}</math> is [[barreled space|barreled]], or else (2) {{mvar|<math>X}}</math> is a [[Baire space]] and {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex. Then every simply bounded subset of {{<math|>L(''X''; ''Y'')}}</math> is equicontinuous.{{sfn | Schaefer|Wolff| 1999 | p=83}}</li>
<li>On an equicontinuous subset {{mvar|<math>H}}</math> of {{<math|>L(''X''; ''Y'')}},</math> the following topologies are identical: (1) topology of pointwise convergence on a total subset of {{mvar|<math>X}}</math>; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.{{sfn | Schaefer|Wolff| 1999 | p=83}}</li>
</ul>
==== Compact convergence {{math|L<sub>c</sub>(''X''; ''Y'')}} ====
By letting <math>\mathcal{{G}</math|𝒢}}> be the set of all compact subsets of {{mvar|<math>X}},</math> {{<math|>L(''X''; ''Y'')}}</math> will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and {{<math|>L(''X''; ''Y'')}}</math> with this topology is denoted by {{math|L<sub>c</sub>(''X''; ''Y'')}}.
The topology of compact convergence on {{<math|>L(''X''; ''Y'')}}</math> has the following properties:
<ul>
<li>If {{mvar|<math>X}}</math> is a [[Fréchet space]] or a [[LF-space]] and if {{mvar|<math>Y}}</math> is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|L<sub>c</sub>(''X''; ''Y'')}} is complete.</li>
<li>On equicontinuous subsets of {{<math|>L(''X''; ''Y'')}},</math> the following topologies coincide:
* The topology of pointwise convergence on a dense subset of {{mvar|<math>X}},</math>
* The topology of pointwise convergence on {{mvar|<math>X}},</math>
* The topology of compact convergence.
* The topology of precompact convergence.</li>
<li>If {{mvar|<math>X}}</math> is a [[Montel space]] and {{mvar|<math>Y}}</math> is a topological vector space, then {{math|L<sub>c</sub>(''X''; ''Y'')}} and {{math|L<sub>b</sub>(''X''; ''Y'')}} have identical topologies.</li>
</ul>
==== Topology of bounded convergence {{math|L<sub>b</sub>(''X''; ''Y'')}} ====
By letting {{<math|𝒢}>\mathcal{G}</math> be the set of all bounded subsets of {{mvar|<math>X}},</math> {{<math|>L(''X''; ''Y'')}}</math> will have '''the topology of bounded convergence on {{mvar|<math>X}}</math>''' or '''the topology of uniform convergence on bounded sets''' and {{<math|>L(''X''; ''Y'')}}</math> with this topology is denoted by {{math|L<sub>b</sub>(''X''; ''Y'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}
The topology of bounded convergence on {{<math|>L(''X''; ''Y'')}}</math> has the following properties:
<ul>
<li>If {{mvar|<math>X}}</math> is a [[bornological space]] and if {{mvar|<math>Y}}</math> is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|L<sub>b</sub>(''X''; ''Y'')}} is complete.</li>
<li>If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are both normed spaces then the topology on {{<math|>L(''X''; ''Y'')}}</math> induced by the usual operator norm is identical to the topology on {{math|L<sub>b</sub>(''X''; ''Y'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}
* In particular, if {{mvar|<math>X}}</math> is a normed space then the usual norm topology on the continuous dual space {{math|''X''{{big|{{'}}}}}} is identical to the topology of bounded convergence on {{math|''X''{{big|{{'}}}}}}.</li>
<li>Every equicontinuous subset of {{<math|>L(''X''; ''Y'')}}</math> is bounded in {{math|L<sub>b</sub>(''X''; ''Y'')}}.</li>
</ul>
{{Main|Polar topology}}
Throughout, we assume that {{mvar|<math>X}}</math> is a TVS.
=== <math>\mathcal{{G}</math|𝒢}}>-topologies versus polar topologies ===
If {{mvar|<math>X}}</math> is a TVS whose [[Bounded set (topological vector space)|bounded]] subsets are exactly the same as its ''weakly'' bounded subsets (e.g. if {{mvar|<math>X}}</math> is a Hausdorff locally convex space), then a <math>\mathcal{{G}</math|𝒢}}>-topology on {{math|''X''{{big|{{'}}}}}} (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a <math>\mathcal{{G}</math|𝒢}}>-topology.
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if {{mvar|<math>X}}</math> is a TVS whose bounded subsets are ''not'' exactly the same as its ''weakly'' bounded subsets, then the notion of "bounded in {{mvar|<math>X}}</math>" is stronger than the notion of "{{math|σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|<math>X}}</math>" (i.e. bounded in {{mvar|<math>X}}</math> implies {{math|σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|<math>X}}</math>) so that a <math>\mathcal{{G}</math|𝒢}}>-topology on {{math|''X''{{big|{{'}}}}}} (as defined in this article) is ''not'' necessarily a polar topology.
One important difference is that polar topologies are always locally convex while <math>\mathcal{{G}</math|𝒢}}>-topologies need not be.
Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article: [[polar topology]].
=== List of polar topologies ===
Suppose that {{mvar|<math>X}}</math> is a TVS whose bounded subsets are the same as its weakly bounded subsets.
:'''Notation''': If {{math|𝛥(''Y'', ''X'')}} denotes a polar topology on {{mvar|<math>Y}}</math> then {{mvar|<math>Y}}</math> endowed with this topology will be denoted by {{math|''Y''<sub>𝛥(''Y'', ''X'')</sub>}} or simply {{math|''Y''<sub>𝛥</sub>}} (e.g. for {{math|σ(''Y'', ''X'')}} we'd have {{math|𝛥 {{=}} σ}} so that {{math|''Y''<sub>σ(''Y'', ''X'')</sub>}} and {{math|''Y''<sub>σ</sub>}} all denote {{mvar|<math>Y}}</math> with endowed with {{math|σ(''Y'', ''X'')}}).
{| class="wikitable"
! Alternative name
|-
| finite subsets of {{mvar|<math>X}}</math>
| {{math|σ(''Y'', ''X'')}}<br/>{{math|s(''Y'', ''X'')}}
| pointwise/simple convergence
|}
== {{math|𝒢-ℋ}}-topologies on spaces of bilinear maps ==
We will let {{math|ℬ(''X'', ''Y''; ''Z'')}} denote the space of separately continuous bilinear maps and {{math|B(''X'', ''Y''; ''Z'')}} denote the space of continuous bilinear maps, where {{mvar|<math>X}}, {{mvar|Y}},</math> and {{mvar|<math>Z}}</math> are topological vector space over the same field (either the real or complex numbers).
In an analogous way to how we placed a topology on {{<math|>L(''X''; ''Y'')}}</math> we can place a topology on {{math|ℬ(''X'', ''Y''; ''Z'')}} and {{math|B(''X'', ''Y''; ''Z'')}}.
Let <math>\mathcal{{G}</math|𝒢}}> (resp. <math>\mathcal{{H}</math|ℋ}}>) be a family of subsets of {{mvar|<math>X}}</math> (resp. {{mvar|<math>Y}}</math>) containing at least one non-empty set.
Let {{<math|𝒢>\mathcal{G} ×\times ℋ}\mathcal{H}</math> denote the collection of all sets {{<math|''>G'' ×\times ''H''}}</math> where {{<math|''>G'' ∈\in 𝒢}\mathcal{G},</math> {{<math|''>H'' ∈\in ℋ}\mathcal{H}.</math>
We can place on {{math|''Z''<sup>''X'' × ''Y''</sup>}} the {{<math|𝒢>\mathcal{G} ×\times ℋ}\mathcal{H}</math>-topology, and consequently on any of its subsets, in particular on {{math|B(''X'', ''Y''; ''Z'')}} and on {{math|ℬ(''X'', ''Y''; ''Z'')}}.
This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math|𝒢-ℋ}}>-topology''' or as the '''topology of uniform convergence on the products {{<math|''>G'' ×\times ''H''}}</math> of <math>\mathcal{{math|𝒢G} ×\times ℋ}\mathcal{H}</math>'''.
However, as before, this topology is not necessarily compatible with the vector space structure of {{math|ℬ(''X'', ''Y''; ''Z'')}} or of {{math|B(''X'', ''Y''; ''Z'')}} without the additional requirement that for all bilinear maps, {{mvar|<math>b}}</math> in this space (that is, in {{math|ℬ(''X'', ''Y''; ''Z'')}} or in {{math|B(''X'', ''Y''; ''Z'')}}) and for all {{<math|''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{<math|''>H'' ∈\in ℋ}\mathcal{H},</math> the set {{math|b(''G'', ''H'')}} is bounded in {{mvar|<math>X}}.</math>
If both <math>\mathcal{{G}</math|𝒢}}> and <math>\mathcal{{H}</math|ℋ}}> consist of bounded sets then this requirement is automatically satisfied if we are topologizing {{math|B(''X'', ''Y''; ''Z'')}} but this may not be the case if we are trying to topologize {{math|ℬ(''X'', ''Y''; ''Z'')}}.
The {{<math|𝒢>\mathcal{G}-ℋ}\mathcal{H}</math>-topology on {{math|ℬ(''X'', ''Y''; ''Z'')}} will be compatible with the vector space structure of {{math|ℬ(''X'', ''Y''; ''Z'')}} if both <math>\mathcal{{G}</math|𝒢}}> and <math>\mathcal{{H}</math|ℋ}}> consists of bounded sets and any of the following conditions hold:
* {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are barrelled spaces and {{mvar|<math>Z}}</math> is locally convex.
* {{mvar|<math>X}}</math> is a [[F-space]], {{mvar|<math>Y}}</math> is metrizable, and {{mvar|<math>Z}}</math> is Hausdorff, in which case {{math|1=ℬ(''X'', ''Y''; ''Z'') = B(''X'', ''Y''; ''Z'')}}.
* {{mvar|<math>X}}, {{mvar|Y}},</math> and {{mvar|<math>Z}}</math> are the strong duals of reflexive Fréchet spaces.
* {{mvar|<math>X}}</math> is normed and {{mvar|<math>Y}}</math> and {{mvar|<math>Z}}</math> the strong duals of reflexive Fréchet spaces.
=== The ε-topology ===
{{Main|Injective tensor product}}
Suppose that {{mvar|<math>X}}, {{mvar|Y}},</math> and {{mvar|<math>Z}}</math> are locally convex spaces and let {{math|𝒢{{'}}}} and {{math|ℋ{{'}}}} be the collections of equicontinuous subsets of {{math|''X''{{big|{{'}}}}}} and {{math|''Y''{{big|{{'}}}}}}, respectively.
Then the {{math|𝒢{{'}}-ℋ{{'}}}}-topology on <math>\mathcal{B}\left( X^{\prime}_{b\left( X^{\prime}, X \right)}, Y^{\prime}_{b\left( X^{\prime}, X \right)}; Z \right)</math> will be a topological vector space topology.
This topology is called the ε-topology and <math>\mathcal{B}\left( X^{\prime}_{b\left( X^{\prime}, X \right)}, Y_{b\left( X^{\prime}, X \right)}; Z \right)</math> with this topology it is denoted by <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{b\left( X^{\prime}, X \right)}, Y^{\prime}_{b\left( X^{\prime}, X \right)}; Z \right)</math> or simply by <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{b}, Y^{\prime}_{b}; Z \right).</math>.
Part of the importance of this vector space and this topology is that it contains many subspace, such as <math>\mathcal{B}\left( X^{\prime}_{\sigma\left( X^{\prime}, X \right)}, Y^{\prime}_{\sigma\left( X^{\prime}, X \right)}; Z \right),</math>, which we denote by <math>\mathcal{B}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}; Z \right).</math>.
When this subspace is given the subspace topology of <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{b}, Y^{\prime}_{b}; Z \right)</math> it is denoted by <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}; Z \right).</math>.
In the instance where {{mvar|<math>Z}}</math> is the field of these vector spaces, <math>\mathcal{B}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math> is a [[tensor product]] of {{mvar|<math>X}}</math> and {{mvar|<math>Y}}.</math>
In fact, if {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex Hausdorff spaces then <math>\mathcal{B}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math> is vector space-isomorphic to <math>L\left( X^{\prime}_{\sigma\left( X^{\prime}, X \right)}; Y_{\sigma(Y^{\prime}, Y)} \right),</math>, which is in turn is equal to <math>L\left( X^{\prime}_{\tau\left( X^{\prime}, X \right)}; Y \right).</math>.
These spaces have the following properties:
* If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are locally convex Hausdorff spaces then {{<math|ℬ<sub>ε</sub>\mathcal{B}_{\varepsilon}<math>\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math> is complete if and only if both {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are complete.
* If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are both normed (or both Banach) then so is <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math>
== See also ==
==Bibliography==
* {{Jarchow Locally Convex Spaces}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{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}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer|Wolff| 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->
{{Functional Analysis}}
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