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In [[mathematics]], a [[linear map]] is a [[function (mathematics)|mapping]] {{math|1=''X'' → ''Y''}} between two [[Module (mathematics)|module]]s (including [[vector space]]s) that preserves the operations of addition and [[scalar (mathematics)|scalar]] multiplication.
By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like [[topology|topologies]] or [[Bornological space|bornologies]], then one can study the subspace of linear maps that preserve this structure.
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Throughout we assume the following:
<ol>
<li>{{mvar|T}} is any non-empty set and {{math|1=𝒢}} is a non-empty collection of subsets of {{mvar|T}} [[Directed set|directed]] by subset inclusion (i.e. for any {{math|1=''G'', ''H'' ∈ 𝒢}} there exists some {{math|1=''K'' ∈ 𝒢}} such that {{math|1=''G'' ∪ ''H'' ⊆ ''K''}}).</li>
<li>{{mvar|Y}} is a [[topological vector space]] (not necessarily Hausdorff or locally convex) and {{math|1=𝒩}} is a basis of neighborhoods of 0 in {{mvar|Y}}.</li>
<li>{{math|1=''Y''<sup>''T''</sup>}} denotes the set of all {{mvar|Y}}-valued functions with ___domain {{mvar|T}}.</li> <li>{{mvar|F}} is a vector subspace of {{math|1=''Y''<sup>''T''</sup>}} (not necessarily consisting of linear maps).</li>
</ol>
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=== Basic neighborhoods at the origin ===
Henceforth assume that {{math|1=''G'' ∈ 𝒢}} and {{math|1=''N'' ∈ 𝒩}}.
;Properties
<ul>
<li>{{math|1=𝒰(''G'', ''N'')}} is an [[Absorbing set|absorbing]] subset of {{mvar|F}} if and only if for all {{math|1=''f'' ∈ ''F''}}, {{mvar|N}} absorbs {{math|1=''f'' (''G'')}}.{{sfn | Narici | 2011 | pp=371-423}}</li>
<li>If {{mvar|N}} is [[Balanced set|balanced]] then so is {{math|1=𝒰(''G'', ''N'')}}.{{sfn | Narici | 2011 | pp=371-423}}</li>
<li>If {{mvar|N}} is [[Convex set|convex]] then so is {{math|1=𝒰(''G'', ''N'')}}.</li>
</ul>
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<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 | 2011 | pp=371-423}}</li>
<li>{{math|1=𝒰(''G'' ∪ ''H'', ''M'' ∩ ''N'') ⊆ 𝒰(''G'', ''M'') ∩ 𝒰(''H'', ''N'')}} for any subsets {{mvar|G}} and {{mvar|H}} of {{mvar|X}} and non-empty subsets {{mvar|M}} and {{mvar|N}} of {{mvar|Y}}.{{sfn | Jarchow | 1981 | pp=43-55}} Thus:
<ul>
<li>If {{math|1=''M'' ⊆ ''N''}} then {{math|1=𝒰(''G'', ''M'') ⊆ 𝒰(''G'', ''N'')}}.{{sfn | Narici | 2011 | pp=371-423}}</li>
<li>If {{math|1=''G'' ⊆ ''H''}} then {{math|1=𝒰(''H'', ''N'') ⊆ 𝒰(''G'', ''N'')}}.</li>
<li>For any {{math|1=''M'', ''N'' ∈ 𝒩}} and subsets {{math|1=''G'', ''H'', ''K''}} of {{mvar|T}}, if {{math|1=''G'' ∪ ''H'' ⊆ ''K''}} then {{math|1=𝒰(''K'', ''M'' ∩ ''N'') ⊆ 𝒰(''G'', ''M'') ∩ 𝒰(''H'', ''N'')}}.</li>
</ul>
</li>
<li>{{math|1=𝒰(∅, ''N'') = ''F''}}.</li>
<li>{{math|1=𝒰(''G'', ''N'') - 𝒰(''G'', ''N'') ⊆ 𝒰(''G'', ''N'' - ''N'')}}.{{sfn | Narici | 2011 | pp=19-45}}</li>
<li>{{math|1=𝒰(''G'', ''M'') + 𝒰(''G'', ''N'') ⊆ 𝒰(''G'', ''M'' + ''N'')}}.{{sfn | Jarchow | 1981 | pp=43-55}}</li>
<li>For any family {{math|1=𝒮}} of subsets of {{mvar|T}}, {{math|1=𝒰({{underset|S ∈ 𝒮|{{big|∪}}}} ''S'', ''N'') = {{underset|S ∈ 𝒮|{{big|∩}}}} 𝒰(''S'', ''N'')}}.{{sfn | Narici | 2011 | pp=19-45}}</li>
<li>For any family {{math|1=ℳ}} of neighborhoods of 0 in {{mvar|Y}}, {{math|1=𝒰(''G'', {{underset|M ∈ ℳ|{{big|∩}}}} ''M'') = {{underset|M ∈ ℳ|{{big|∩}}}} 𝒰(''G'', ''M'')}}.{{sfn | Narici | 2011 | pp=19-45}}</li>
</ul>
=== {{math|1=𝒢}}-topology ===
Then the set {{math|1={𝒰(''G'', ''N'') : ''G'' ∈ 𝒢, ''N'' ∈ 𝒩}}} forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set {{math|1=𝒰(''G'', ''N'')}} 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|F}}, where this topology is ''not'' necessarily a vector topology (i.e. it might not make {{mvar|F}} into a TVS).
This topology does not depend on the neighborhood basis {{math|1=𝒩}} that was chosen and it is known as the '''topology of uniform convergence on the sets in {{math|1=𝒢}}''' or as the '''{{math|1=𝒢}}-topology'''.{{sfn | Schaefer | 1999 | pp=79-88}}
However, this name is frequently changed according to the types of sets that make up {{math|1=𝒢}} (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|1=𝒢}} 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|1=𝒢}} is the collection of compact subsets of {{mvar|T}} (and {{mvar|T}} is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of {{mvar|T}}.</ref>).
A subset {{math|1=𝒢<sub>1</sub>}} of {{math|1=𝒢}} is said to be '''fundamental with respect to {{math|1=𝒢}}''' if each {{math|1=''G'' ∈ 𝒢}} is a subset of some element in {{math|1=𝒢<sub>1</sub>}}.
In this case, the collection {{math|1=𝒢}} can be replaced by {{math|1=𝒢<sub>1</sub>}} without changing the topology on {{mvar|F}}.{{sfn | Schaefer | 1999 | pp=79-88}}
One may also replace {{math|1=𝒢}} with the collection of all subsets of all finite unions of elements of {{math|1=𝒢}} without changing the resulting {{math|1=𝒢}}-topology on {{mvar|F}}.{{sfn | Narici | 2011 | pp=19-45}}
:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Call a subset {{mvar|B}} of {{mvar|T}} '''{{mvar|F}}-bounded''' if {{math|1=''f'' (''B'')}} is a bounded subset of {{mvar|Y}} for every {{math|1=''f'' ∈ ''F''}}.
{{Math theorem|name=Theorem{{sfn | Schaefer | 1999 | pp=79-88}}{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=
The {{math|1=𝒢}}-topology on {{mvar|F}} is compatible with the vector space structure of {{mvar|F}} if and only if every {{math|1=''G'' ∈ 𝒢}} is {{mvar|F}}-bounded;
that is, if and only if for every {{math|1=''G'' ∈ 𝒢}} and every {{math|1=''f'' ∈ ''F''}}, {{math|1=''f'' (''G'')}} is [[Bounded set (topological vector space)|bounded]] in {{mvar|Y}}.
}}
==== Nets and uniform convergence ====
:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Let {{math|1=''f'' ∈ ''F''}} and let {{math|1=''f''<sub>•</sub> = (''f''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} be a [[Net (mathematics)|net]] in {{mvar|F}}. Then for any subset {{mvar|G}} of {{mvar|T}}, say that {{math|1=''f''<sub>•</sub>}} '''converges uniformly to {{mvar|f}} on {{mvar|G}}''' if for every {{math|1=''N'' ∈ 𝒩}} there exists some {{math|1=''i''<sub>0</sub> ∈ ''I''}} such that for every {{math|1=''i'' ∈ ''I''}} satisfying {{math|1=''i'' ≥ ''i''<sub>0</sub>}}, {{math|1=''f''<sub>''i''</sub> - ''f'' ∈ 𝒰(''G'', ''N'')}} (or equivalently, {{math|1=''f''<sub>''i''</sub>(''g'') - ''f'' (''g'') ∈ ''N''}} for every {{math|1=''g'' ∈ ''G''}}).
{{Math theorem|name=Theorem{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=
If {{math|1=''f'' ∈ ''F''}} and if {{math|1=''f''<sub>•</sub> = (''f''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} is a net in {{mvar|F}}, then {{math|1=''f''<sub>•</sub> → ''f''}} in the {{math|1=𝒢}}-topology on {{mvar|F}} if and only if for every {{math|1=''G'' ∈ 𝒢}}, {{math|1=''f''<sub>•</sub>}} converges uniformly to {{mvar|f}} on {{mvar|G}}.
}}
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;Local convexity
If {{mvar|Y}} is [[locally convex]] then so is the {{math|1=𝒢}}-topology on {{mvar|F}} and if {{math|1=(''p''<sub>''i''</sub>)<sub>''i'' ∈ ''I''</sub>}} is a family of continuous seminorms generating this topology on {{mvar|Y}} then the {{math|1=𝒢}}-topology is induced by the following family of seminorms:
:{{math|1=''p''<sub>''G'',''i''</sub>( ''f'' ) {{=}}}} {{underset|{{math|1=''x'' ∈ ''G''}}|sup}} {{math|1=''p''<sub>''i''</sub>( ''f''(''x''))}},
as {{mvar|G}} varies over {{math|1=𝒢}} and {{mvar|i}} varies over {{mvar|I}}.{{sfn | Schaefer | 1999 | p=81}}
;Hausdorffness
If {{mvar|Y}} is [[Hausdorff space|Hausdorff]] and {{math|1=''T'' = {{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} then the {{math|1=𝒢}}-topology on {{mvar|F}} is Hausdorff.{{sfn | Jarchow | 1981 | pp=43-55}}
Suppose that {{mvar|T}} is a topological space.
If {{mvar|Y}} is [[Hausdorff space|Hausdorff]] and {{mvar|F}} is the vector subspace of {{math|1=''Y''<sup>''T''</sup>}} consisting of all continuous maps that are bounded on every {{math|1=''G'' ∈ 𝒢}} and if {{math|1={{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|T}} then the {{math|1=𝒢}}-topology on {{mvar|F}} is Hausdorff.
;Boundedness
A subset {{mvar|H}} of {{mvar|F}} is [[Bounded set (topological vector space)|bounded]] in the {{math|1=𝒢}}-topology if and only if for every {{math|1=''G'' ∈ 𝒢}}, {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|Y}}.{{sfn | Schaefer | 1999 | p=81}}
=== Examples of 𝒢-topologies ===
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;Pointwise convergence
If we let {{math|1=𝒢}} be the set of all finite subsets of {{mvar|T}} then the {{math|1=𝒢}}-topology on {{mvar|F}} is called the '''topology of pointwise convergence'''.
The topology of pointwise convergence on {{mvar|F}} is identical to the subspace topology that {{mvar|F}} inherits from {{math|1=''Y''<sup>''T''</sup>}} when {{math|1=''Y''<sup>''T''</sup>}} is endowed with the usual [[product topology]].
If {{mvar|X}} is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and {{math|1=C(''X'')}} is the space of all real (or complex) valued continuous functions on {{mvar|X}}, the topology of pointwise convergence on {{math|1=C(''X'')}} is [[Metrizable TVS|metrizable]] if and only if {{mvar|X}} is countable.{{sfn | Jarchow | 1981 | pp=43-55}}
== 𝒢-topologies on spaces of continuous linear maps ==
Throughout this section we will assume that {{mvar|X}} and {{mvar|Y}} are [[topological vector space]]s.
{{math|1=𝒢}} will be a non-empty collection of subsets of {{mvar|X}} [[Directed set|directed]] by inclusion.
:'''Notation''': {{math|1=L(''X''; ''Y'')}} will denote the vector space of all continuous linear maps from {{mvar|X}} into {{mvar|Y}}. If {{math|1=L(''X''; ''Y'')}} is given the {{math|1=𝒢}}-topology inherited from {{math|1=''Y''<sup>''X''</sup>}} then this space with this topology is denoted by {{math|1=L<sub>𝒢</sub>(''X'', ''Y'')}}.
:'''Notation''': The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space {{mvar|X}} over the field {{math|1=𝔽}} (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space {{math|1=L(''X''; 𝔽)}} and is denoted by {{math|1=''X''{{big|{{'}}}}}}.
The {{math|1=𝒢}}-topology on {{math|1=L(''X''; ''Y'')}} is compatible with the vector space structure of {{math|1=L(''X''; ''Y'')}} if and only if for all {{math|1=''G'' ∈ 𝒢}} and all {{math|1=''f'' ∈ L(''X''; ''Y'')}} the set {{math|1=''f''(''G'')}} is bounded in {{mvar|Y}}, which we will assume to be the case for the rest of the article.
Note in particular that this is the case if {{math|1=𝒢}} consists of [[Bounded set (topological vector space)|(von-Neumann) bounded]] subsets of {{mvar|X}}.
=== Assumptions on 𝒢 ===
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;Assumptions that guarantee a vector topology
:'''Assumption''' ({{math|1=𝒢}} is directed): {{math|1=𝒢}} will be a non-empty collection of subsets of {{mvar|X}} [[Directed set|directed]] by (subset) inclusion. That is, for any {{math|1=''G'', ''H'' ∈ 𝒢}}, there exists {{math|1=''K'' ∈ 𝒢}} such that {{math|1=''G'' ∪ ''H'' ⊆ ''K''}}.
The above assumption guarantees that the collection of sets {{math|1=𝒰(''G'', ''N'')}} forms a [[filter base]].
The next assumption will guarantee that the sets {{math|1=𝒰(''G'', ''N'')}} are [[Balanced set|balanced]].
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdonsome.
:'''Assumption''' ({{math|1=''N'' ∈ 𝒩}} are balanced): {{math|1=𝒩}} is a neighborhoods basis of 0 in {{mvar|Y}} that consists entirely of [[Balanced set|balanced]] sets.
The following assumption is very commonly made because it will guarantee that each set {{math|1=𝒰(''G'', ''N'')}} is absorbing in {{math|1=L(''X''; ''Y'')}}.
:'''Assumption''' ({{math|1=''G'' ∈ 𝒢}} are bounded): {{math|1=𝒢}} is assumed to consist entirely of bounded subsets of {{mvar|X}}.
;Other possible assumptions
The next theorem gives ways in which {{math|1=𝒢}} can be modified without changing the resulting {{math|1=𝒢}}-topology on {{mvar|Y}}.
{{Math theorem|name=Theorem{{sfn | Narici | 2011 | pp=371-423}}|math_statement=
Let {{math|1=𝒢}} be a non-empty collection of bounded subsets of {{mvar|X}}. Then the {{math|1=𝒢}}-topology on {{math|1=L(''X''; ''Y'')}} is not altered if {{math|1=𝒢}} is replaced by any of the following collections of (also bounded) subsets of {{mvar|X}}:
<ol>
<li>all subsets of all finite unions of sets in {{math|1=𝒢}};</li>
<li>all scalar multiples of all sets in {{math|1=𝒢}};</li>
<li>all finite [[Minkowski sum]]s of sets in {{math|1=𝒢}};</li>
<li>the [[Balanced set|balanced hull]] of every set in {{math|1=𝒢}};</li>
<li>the closure of every set in {{math|1=𝒢}};</li>
</ol>
and if {{mvar|X}} and {{mvar|Y}} 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|1=𝒢}}.</li>
</ol>
}}
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;Common assumptions
Some authors (e.g. Narici) require that {{math|1=𝒢}} satisfy the following condition, which implies, in particular, that {{math|1=𝒢}} is [[Directed set|directed]] by subset inclusion:
:{{math|1=𝒢}} is assumed to be closed with respect to the formation of subsets of finite unions of sets in {{math|1=𝒢}} (i.e. every subset of every finite union of sets in {{math|1=𝒢}} belongs to {{math|1=𝒢}}).
Some authors (e.g. Trèves) require that {{math|1=𝒢}} be directed under subset inclusion and that it satisfy the following condition:
:If {{math|1=''G'' ∈ 𝒢}} and {{mvar|s}} is a scalar then there exists a {{math|1=''H'' ∈ 𝒢}} such that {{math|1=''sG'' ⊆ ''H''}}.
If {{math|1=𝒢}} is a [[bornology]] on {{mvar|X}}, which is often the case, then these axioms are satisfied.
If {{math|1=𝒢}} is a [[saturated family]] of [[Bounded set (topological vector space)|bounded]] subsets of {{mvar|X}} then these axioms are also satisfied.
=== Properties ===
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;Hausdorffness
:'''Definition''':{{sfn | Schaefer | 1999 | p=80}} If {{mvar|T}} is a TVS then we say that {{math|1=𝒢}} is '''total in {{mvar|T}}''' if the [[linear span]] of {{math|1={{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|T}}.
If {{mvar|F}} is the vector subspace of {{math|1=''Y''<sup>''T''</sup>}} consisting of all continuous linear maps that are bounded on every {{math|1=''G'' ∈ 𝒢}}, then the {{math|1=𝒢}}-topology on {{mvar|F}} is Hausdorff if {{mvar|Y}} is Hausdorff and {{math|1=𝒢}} is total in {{mvar|T}}.{{sfn | Narici | 2011 | pp=371-423}}
;Completeness
For the following theorems, suppose that {{mvar|X}} is a topological vector space and {{mvar|Y}} is a [[locally convex]] Hausdorff spaces and {{math|1=𝒢}} is a collection of bounded subsets of {{mvar|X}} that covers {{mvar|X}}, is directed by subset inclusion, and satisfies the following condition: if {{math|1=''G'' ∈ 𝒢}} and {{mvar|s}} is a scalar then there exists a {{math|1=''H'' ∈ 𝒢}} such that {{math|1=''sG'' ⊆ ''H''}}.
<ul>
<li>{{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}} is complete if
{{ordered list|
| {{mvar|X}} is locally convex and Hausdorff,
| {{mvar|Y}} is complete, and
| whenever {{math|1=''u'' : ''X'' → ''Y''}} is a linear map then {{mvar|u}} restricted to every set {{math|1=''G'' ∈ 𝒢}} is continuous implies that {{mvar|u}} is continuous,
}}</li>
<li>If {{mvar|X}} is a Mackey space then {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}}is complete if and only if both <math>X^{\prime}_{\mathcal{G}}</math> and {{mvar|Y}} are complete.</li>
<li>If {{mvar|X}} is [[Barrelled space|barrelled]] then {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}} is Hausdorff and [[quasi-complete]].</li>
<li>Let {{mvar|X}} and {{mvar|Y}} be TVSs with {{mvar|Y}} [[quasi-complete]] and assume that (1) {{mvar|X}} is [[barreled space|barreled]], or else (2) {{mvar|X}} is a [[Baire space]] and {{mvar|X}} and {{mvar|Y}} are locally convex. If {{math|1=𝒢}} covers {{mvar|X}} then every closed equicontinuous subset of {{math|1=L(''X''; ''Y'')}} is complete in {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}} and {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}} is quasi-complete.{{sfn | Schaefer | 1999 | p=83}}</li>
<li>Let {{mvar|X}} be a [[bornological space]], {{mvar|Y}} a locally convex space, and {{math|1=𝒢}} a family of bounded subsets of {{mvar|X}} such that the range of every null sequence in {{mvar|X}} is contained in some {{math|1=''G'' ∈ 𝒢}}. If {{mvar|Y}} is [[quasi-complete]] (resp. complete) then so is {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}}.{{sfn | Schaefer | 1999 | p=117}}</li>
</ul>
;Boundedness
Let {{mvar|X}} and {{mvar|Y}} be topological vector spaces and {{mvar|H}} be a subset of {{math|1=L(''X''; ''Y'')}}.
Then the following are equivalent:{{sfn | Schaefer | 1999 | p=81}}
<ol>
<li>{{mvar|H}} is [[Bounded set (topological vector space)|bounded]] in {{math|1=L<sub>𝒢</sub>(''X''; ''Y'')}};</li>
<li>For every {{math|1=''G'' ∈ 𝒢}}, {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|Y}};{{sfn | Schaefer | 1999 | p=81}}</li>
<li>For every neighborhood {{mvar|V}} of 0 in {{mvar|Y}} the set {{math|1={{underset|''h'' ∈ ''H''|{{big|∩}}}} ''h'' <sup>-1</sup>(''V'')}} [[Absorbing set|absorbs]] every {{math|1=''G'' ∈ 𝒢}}.</li>
</ol>
Furthermore,
<ul>
<li>If {{mvar|X}} and {{mvar|Y}} are locally convex Hausdorff space and if {{mvar|H}} is bounded in {{math|1=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|X}}.{{sfn | Schaefer | 1999 | p=82}}</li>
<li>If {{mvar|X}} and {{mvar|Y}} are locally convex Hausdorff spaces and if {{mvar|X}} is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of {{math|1=L(''X''; ''Y'')}} are identical for all {{math|1=𝒢}}-topologies where {{math|1=𝒢}} is any family of bounded subsets of {{mvar|X}} covering {{mvar|X}}.{{sfn | Schaefer | 1999 | p=82}}</li>
<li>If {{math|1=𝒢}} is any collection of bounded subsets of {{mvar|X}} whose union is total in {{mvar|X}} then every equicontinuous subset of {{math|1=L(''X''; ''Y'')}} is bounded in the {{math|1=𝒢}}-topology.{{sfn | Schaefer | 1999 | p=83}}</li>
</ul>
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{| class="wikitable"
|-
! {{math|1=𝒢 ⊆ 𝒫(''X'')}} ("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
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|-
| finite subsets of {{mvar|X}}
| {{math|1=L<sub>σ</sub>(''X''; ''Y'')}}
| pointwise/simple convergence
| topology of simple convergence
Line 218:
|-
| compact convex subsets of {{mvar|X}}
| {{math|1=L<sub>γ</sub>(''X''; ''Y'')}}
| compact convex convergence
|
|-
| compact subsets of {{mvar|X}}
| {{math|1=L<sub>c</sub>(''X''; ''Y'')}}
| compact convergence
|
|-
| bounded subsets of {{mvar|X}}
| {{math|1=L<sub>b</sub>(''X''; ''Y'')}}
| bounded convergence
| strong topology
|}
==== The topology of pointwise convergence {{math|1=L<sub>σ</sub>(''X''; ''Y'')}} ====
By letting {{math|1=𝒢}} be the set of all finite subsets of {{mvar|X}}, {{math|1=L(''X''; ''Y'')}} will have the '''weak topology on {{math|1=L(''X''; ''Y'')}}''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and {{math|1=L(''X''; ''Y'')}} with this topology is denoted by {{math|1=L<sub>𝜎</sub>(''X''; ''Y'')}}.
Unfortunately, this topology is also sometimes called '''the strong operator topology''', which may lead to ambiguity;{{sfn | Narici | 2011 | pp=371-423}} for this reason, this article will avoid referring to this topology by this name.
:'''Definition''': A subset of {{math|1=L(''X''; ''Y'')}} is called '''simply bounded''' or '''weakly bounded''' if it is bounded in {{math|1=L<sub>𝜎</sub>(''X''; ''Y'')}}.
The weak-topology on {{math|1=L(''X''; ''Y'')}} has the following properties:
<ul>
<li>If {{mvar|X}} is [[Separable space|separable]] (i.e. has a countable dense subset) and if {{mvar|Y}} is a metrizable topological vector space then every equicontinuous subset {{mvar|H}} of {{math|1=L<sub>𝜎</sub>(''X''; ''Y'')}} is metrizable; if in addition {{mvar|Y}} is separable then so is {{mvar|H}}.{{sfn | Schaefer | 1999 | p=87}}
* So in particular, on every equicontinuous subset of {{math|1=L(''X''; ''Y'')}}, the topology of pointwise convergence is metrizable.</li>
<li>Let {{math|1=''Y''<sup>''X''</sup>}} denote the space of all functions from {{mvar|X}} into {{mvar|Y}}. If {{math|1=L(''X''; ''Y'')}} is given the topology of pointwise convergence then space of all linear maps (continuous or not) {{mvar|X}} into {{mvar|Y}} is closed in {{math|1=''Y''<sup>''X''</sup>}}.
* In addition, {{math|1=L(''X''; ''Y'')}} is dense in the space of all linear maps (continuous or not) {{mvar|X}} into {{mvar|Y}}.</li>
<li>Suppose {{mvar|X}} and {{mvar|Y}} are locally convex. Any simply bounded subset of {{math|1=L(''X''; ''Y'')}} is bounded when {{math|1=L(''X''; ''Y'')}} has the topology of uniform convergence on convex, [[balanced set|balanced]], bounded, complete subsets of {{mvar|X}}. If in addition {{mvar|X}} is [[quasi-complete]] then the families of bounded subsets of {{math|1=L(''X''; ''Y'')}} are identical for all {{math|1=𝒢}}-topologies on {{math|1=L(''X''; ''Y'')}} such that {{math|1=𝒢}} is a family of bounded sets covering {{mvar|X}}.{{sfn | Schaefer | 1999 | p=82}}</li>
</ul>
;Equicontinuous subsets
<ul>
<li>The weak-closure of an equicontinuous subset of {{math|1=L(''X''; ''Y'')}} is equicontinuous.</li>
<li>If {{mvar|Y}} is locally convex, then the convex balanced hull of an equicontinuous subset of {{math|1=L(''X''; ''Y'')}} is equicontinuous.</li>
<li>Let {{mvar|X}} and {{mvar|Y}} be TVSs and assume that (1) {{mvar|X}} is [[barreled space|barreled]], or else (2) {{mvar|X}} is a [[Baire space]] and {{mvar|X}} and {{mvar|Y}} are locally convex. Then every simply bounded subset of {{math|1=L(''X''; ''Y'')}} is equicontinuous.{{sfn | Schaefer | 1999 | p=83}}</li>
<li>On an equicontinuous subset {{mvar|H}} of {{math|1=L(''X''; ''Y'')}}, the following topologies are identical: (1) topology of pointwise convergence on a total subset of {{mvar|X}}; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.{{sfn | Schaefer | 1999 | p=83}}</li>
</ul>
==== Compact convergence {{math|1=L<sub>c</sub>(''X''; ''Y'')}} ====
By letting {{math|1=𝒢}} be the set of all compact subsets of {{mvar|X}}, {{math|1=L(''X''; ''Y'')}} will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and {{math|1=L(''X''; ''Y'')}} with this topology is denoted by {{math|1=L<sub>c</sub>(''X''; ''Y'')}}.
The topology of compact convergence on {{math|1=L(''X''; ''Y'')}} has the following properties:
<ul>
<li>If {{mvar|X}} is a [[Fréchet space]] or a [[LF-space]] and if {{mvar|Y}} is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|1=L<sub>c</sub>(''X''; ''Y'')}} is complete.</li>
<li>On equicontinuous subsets of {{math|1=L(''X''; ''Y'')}}, the following topologies coincide:
* The topology of pointwise convergence on a dense subset of {{mvar|X}},
* The topology of pointwise convergence on {{mvar|X}},
* The topology of compact convergence.
* The topology of precompact convergence.</li>
<li>If {{mvar|X}} is a [[Montel space]] and {{mvar|Y}} is a topological vector space, then {{math|1=L<sub>c</sub>(''X''; ''Y'')}} and {{math|1=L<sub>b</sub>(''X''; ''Y'')}} have identical topologies.</li>
</ul>
==== Topology of bounded convergence {{math|1=L<sub>b</sub>(''X''; ''Y'')}} ====
By letting {{math|1=𝒢}} be the set of all bounded subsets of {{mvar|X}}, {{math|1=L(''X''; ''Y'')}} will have '''the topology of bounded convergence on {{mvar|X}}''' or '''the topology of uniform convergence on bounded sets''' and {{math|1=L(''X''; ''Y'')}} with this topology is denoted by {{math|1=L<sub>b</sub>(''X''; ''Y'')}}.{{sfn | Narici | 2011 | pp=371-423}}
The topology of bounded convergence on {{math|1=L(''X''; ''Y'')}} has the following properties:
<ul>
<li>If {{mvar|X}} is a [[bornological space]] and if {{mvar|Y}} is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|1=L<sub>b</sub>(''X''; ''Y'')}} is complete.</li>
<li>If {{mvar|X}} and {{mvar|Y}} are both normed spaces then the topology on {{math|1=L(''X''; ''Y'')}} induced by the usual operator norm is identical to the topology on {{math|1=L<sub>b</sub>(''X''; ''Y'')}}.{{sfn | Narici | 2011 | pp=371-423}}
* In particular, if {{mvar|X}} is a normed space then the usual norm topology on the continuous dual space {{math|1=''X'' {{big|{{'}}}}}} is identical to the topology of bounded convergence on {{math|1=''X'' {{big|{{'}}}}}}.</li>
<li>Every equicontinuous subset of {{math|1=L(''X''; ''Y'')}} is bounded in {{math|1=L<sub>b</sub>(''X''; ''Y'')}}.</li>
</ul>
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Throughout, we assume that {{mvar|X}} is a TVS.
=== {{math|1=𝒢}}-topologies versus polar topologies ===
If {{mvar|X}} is a TVS whose [[Bounded set (topological vector space)|bounded]] subsets are exactly the same as its ''weakly'' bounded subsets (e.g. if {{mvar|X}} is a Hausdorff locally convex space), then a {{math|1=𝒢}}-topology on {{math|1=''X''{{big|{{'}}}}}} (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a {{math|1=𝒢}}-topology.
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if {{mvar|X}} is a TVS whose bounded subsets are ''not'' exactly the same as its ''weakly'' bounded subsets, then the notion of "bounded in {{mvar|X}}" is stronger than the notion of "{{math|1=σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|X}}" (i.e. bounded in {{mvar|X}} implies {{math|1=σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|X}}) so that a {{math|1=𝒢}}-topology on {{math|1=''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|1=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]].
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Suppose that {{mvar|X}} is a TVS whose bounded subsets are the same as its weakly bounded subsets.
:'''Notation''': If {{math|1=𝛥(''Y'', ''X'')}} denotes a polar topology on {{mvar|Y}} then {{mvar|Y}} endowed with this topology will be denoted by {{math|1=''Y''<sub>𝛥(''Y'', ''X'')</sub>}} or simply {{math|1=''Y''<sub>𝛥</sub>}} (e.g. for {{math|1=σ(''Y'', ''X'')}} we'd have {{math|1=𝛥 {{=}} σ}} so that {{math|1=''Y''<sub>σ(''Y'', ''X'')</sub>}} and {{math|1=''Y''<sub>σ</sub>}} all denote {{mvar|Y}} with endowed with {{math|1=σ(''Y'', ''X'')}}).
{| class="wikitable"
|-
! {{math|1=𝒢 ⊆ 𝒫(''X'')}}<br/>("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
Line 314:
|-
| finite subsets of {{mvar|X}}
| {{math|1=σ(''Y'', ''X'')}}<br/>{{math|1=s(''Y'', ''X'')}}
| pointwise/simple convergence
| [[Weak topology|weak/weak* topology]]
|-
| {{math|1=σ(''X'', ''Y'')}}-compact [[Absolutely convex set|disk]]s
| {{math|1=τ(''Y'', ''X'')}}
|
| [[Mackey topology]]
|-
| {{math|1=σ(''X'', ''Y'')}}-compact convex subsets
| {{math|1=γ(''Y'', ''X'')}}
| compact convex convergence
|
|-
| {{math|1=σ(''X'', ''Y'')}}-compact subsets<br/>(or balanced {{math|1=σ(''X'', ''Y'')}}-compact subsets)
| {{math|1=c(''Y'', ''X'')}}
| compact convergence
|
|-
| {{math|1=σ(''X'', ''Y'')}}-bounded subsets
| {{math|1=b(''Y'', ''X'')}}<br/>{{math|1=𝛽(''Y'', ''X'')}}
| bounded convergence
| [[Strong dual space|strong topology]]
|}
== {{math|1=𝒢-ℋ}}-topologies on spaces of bilinear maps ==
We will let {{math|1=ℬ(''X'', ''Y''; ''Z'')}} denote the space of separately continuous bilinear maps and {{math|1=B(''X'', ''Y''; ''Z'')}} denote the space of continuous bilinear maps, where {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}} 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|1=L(''X''; ''Y'')}} we can place a topology on {{math|1=ℬ(''X'', ''Y''; ''Z'')}} and {{math|1=B(''X'', ''Y''; ''Z'')}}.
Let {{math|1=𝒢}} (resp. {{math|1=ℋ}}) be a family of subsets of {{mvar|X}} (resp. {{mvar|Y}}) containing at least one non-empty set.
Let {{math|1=𝒢 × ℋ}} denote the collection of all sets {{math|1=''G'' × ''H''}} where {{math|1=''G'' ∈ 𝒢}}, {{math|1=''H'' ∈ ℋ}}.
We can place on {{math|1=''Z''<sup>''X'' × ''Y''</sup>}} the {{math|1=𝒢 × ℋ}}-topology, and consequently on any of its subsets, in particular on {{math|1=B(''X'', ''Y''; ''Z'')}} and on {{math|1=ℬ(''X'', ''Y''; ''Z'')}}.
This topology is known as the '''{{math|1=𝒢-ℋ}}-topology''' or as the '''topology of uniform convergence on the products {{math|1=''G'' × ''H''}} of {{math|1=𝒢 × ℋ}}'''.
However, as before, this topology is not necessarily compatible with the vector space structure of {{math|1=ℬ(''X'', ''Y''; ''Z'')}} or of {{math|1=B(''X'', ''Y''; ''Z'')}} without the additional requirement that for all bilinear maps, {{mvar|b}} in this space (that is, in {{math|1=ℬ(''X'', ''Y''; ''Z'')}} or in {{math|1=B(''X'', ''Y''; ''Z'')}}) and for all {{math|1=''G'' ∈ 𝒢}} and {{math|1=''H'' ∈ ℋ}}, the set {{math|1=b(''G'', ''H'')}} is bounded in {{mvar|X}}.
If both {{math|1=𝒢}} and {{math|1=ℋ}} consist of bounded sets then this requirement is automatically satisfied if we are topologizing {{math|1=B(''X'', ''Y''; ''Z'')}} but this may not be the case if we are trying to topologize {{math|1=ℬ(''X'', ''Y''; ''Z'')}}.
The {{math|1=𝒢-ℋ}}-topology on {{math|1=ℬ(''X'', ''Y''; ''Z'')}} will be compatible with the vector space structure of {{math|1=ℬ(''X'', ''Y''; ''Z'')}} if both {{math|1=𝒢}} and {{math|1=ℋ}} consists of bounded sets and any of the following conditions hold:
* {{mvar|X}} and {{mvar|Y}} are barrelled spaces and {{mvar|Z}} is locally convex.
* {{mvar|X}} is a [[F-space]], {{mvar|Y}} is metrizable, and {{mvar|Z}} is Hausdorff, in which case {{math|1=ℬ(''X'', ''Y''; ''Z'') = B(''X'', ''Y''; ''Z'')}}.
* {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}} are the strong duals of reflexive Fréchet spaces.
* {{mvar|X}} is normed and {{mvar|Y}} and {{mvar|Z}} the strong duals of reflexive Fréchet spaces.
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{{Main|Injective tensor product}}
Suppose that {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}} are locally convex spaces and let {{math|1=𝒢{{'}}}} and {{math|1=ℋ {{'}}}} be the collections of equicontinuous subsets of {{math|1=''X''{{big|{{'}}}}}} and {{math|1=''Y''{{big|{{'}}}}}}, respectively.
Then the {{math|1=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>.
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These spaces have the following properties:
* If {{mvar|X}} and {{mvar|Y}} are locally convex Hausdorff spaces then {{math|1=ℬ<sub>ε</sub>}}<math>\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math> is complete if and only if both {{mvar|X}} and {{mvar|Y}} are complete.
* If {{mvar|X}} and {{mvar|Y}} are both normed (or both Banach) then so is <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math>
| |||