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<ul>
<li>{{math|𝒰(''G'', ''N'')}} is an [[Absorbing set|absorbing]] subset of {{mvar|F}} if and only if for all {{math|''f'' ∈ ''F''}}, {{mvar|N}} absorbs {{math|''f'' (''G'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>If {{mvar|N}} is [[Balanced set|balanced]] then so is {{math|𝒰(''G'', ''N'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}</li>
<li>If {{mvar|N}} is [[Convex set|convex]] then so is {{math|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 | Beckenstein | 2011 | pp=371-423}}</li>
<li>{{math|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|''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|T}}, if {{math|''G'' ∪ ''H'' ⊆ ''K''}} then {{math|𝒰(''K'', ''M'' ∩ ''N'') ⊆ 𝒰(''G'', ''M'') ∩ 𝒰(''H'', ''N'')}}.</li>
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</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|𝒮}} of subsets of {{mvar|T}}, {{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|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 | Beckenstein | 2011 | pp=19-45}}</li>
</ul>
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A subset {{math|𝒢<sub>1</sub>}} of {{math|𝒢}} is said to be '''fundamental with respect to {{math|𝒢}}''' if each {{math|''G'' ∈ 𝒢}} is a subset of some element in {{math|𝒢<sub>1</sub>}}.
In this case, the collection {{math|𝒢}} can be replaced by {{math|𝒢<sub>1</sub>}} without changing the topology on {{mvar|F}}.{{sfn | Schaefer | 1999 | pp=79-88}}
One may also replace {{math|𝒢}} with the collection of all subsets of all finite unions of elements of {{math|𝒢}} without changing the resulting {{math|𝒢}}-topology on {{mvar|F}}.{{sfn | Narici | Beckenstein | 2011 | pp=19-45}}
:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Call a subset {{mvar|B}} of {{mvar|T}} '''{{mvar|F}}-bounded''' if {{math|''f'' (''B'')}} is a bounded subset of {{mvar|Y}} for every {{math|''f'' ∈ ''F''}}.
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The next theorem gives ways in which {{math|𝒢}} can be modified without changing the resulting {{math|𝒢}}-topology on {{mvar|Y}}.
{{Math theorem|name=Theorem{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}|math_statement=
Let {{math|𝒢}} be a non-empty collection of bounded subsets of {{mvar|X}}. Then the {{math|𝒢}}-topology on {{math|L(''X''; ''Y'')}} is not altered if {{math|𝒢}} is replaced by any of the following collections of (also bounded) subsets of {{mvar|X}}:
<ol>
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:'''Definition''':{{sfn | Schaefer | 1999 | p=80}} If {{mvar|T}} is a TVS then we say that {{math|𝒢}} is '''total in {{mvar|T}}''' if the [[linear span]] of {{math|{{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|T}}.
If {{mvar|F}} is the vector subspace of {{math|''Y''<sup>''T''</sup>}} consisting of all continuous linear maps that are bounded on every {{math|''G'' ∈ 𝒢}}, then the {{math|𝒢}}-topology on {{mvar|F}} is Hausdorff if {{mvar|Y}} is Hausdorff and {{math|𝒢}} is total in {{mvar|T}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}
;Completeness
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By letting {{math|𝒢}} be the set of all finite subsets of {{mvar|X}}, {{math|L(''X''; ''Y'')}} will have the '''weak topology on {{math|L(''X''; ''Y'')}}''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and {{math|L(''X''; ''Y'')}} 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'')}} is called '''simply bounded''' or '''weakly bounded''' if it is bounded in {{math|L<sub>𝜎</sub>(''X''; ''Y'')}}.
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==== Topology of bounded convergence {{math|L<sub>b</sub>(''X''; ''Y'')}} ====
By letting {{math|𝒢}} be the set of all bounded subsets of {{mvar|X}}, {{math|L(''X''; ''Y'')}} will have '''the topology of bounded convergence on {{mvar|X}}''' or '''the topology of uniform convergence on bounded sets''' and {{math|L(''X''; ''Y'')}} 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'')}} 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|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|L(''X''; ''Y'')}} 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|X}} 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'')}} is bounded in {{math|L<sub>b</sub>(''X''; ''Y'')}}.</li>
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{{Reflist}}
* {{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 | 1999 | p=}} -->
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