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In [[mathematics]], aparticularly [[linearfunctional mapanalysis]], isspaces aof [[functionlinear (mathematics)|mappingmap]] {{math|1=''X'' → ''Y''}}s between two [[Module (mathematics)|module]]s (including [[vector space]]s) thatcan preservesbe theendowed operationswith ofa additionvariety andof [[scalarTopology (mathematicsstructure)|scalartopologies]]. multiplicationStudying space of linear maps and these topologies can give insight into the spaces themselves.
The article [[operator topologies]] discusses topologies on spaces of linear maps between [[normed space]]s, whereas this article discusses topologies on such spaces in the more general setting of [[topological vector space]]s (TVSs).
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.
== Topologies of uniform convergence on arbitrary spaces of maps ==
Throughout we assume, the following is assumed:
<ol>
<li>{{mvar|<math>T}}</math> is any non-empty set and <math>\mathcal{{G}</math|1=𝒢}}> is a non-empty collection of subsets of {{mvar|<math>T}}</math> [[Directed set|directed]] by subset inclusion (i.e. for any {{<math|1=''>G'', ''H'' ∈\in 𝒢}\mathcal{G}</math> there exists some {{<math|1=''>K'' ∈\in 𝒢}\mathcal{G}</math> such that {{<math|1=''>G'' ∪\cup ''H'' ⊆\subseteq ''K''}}</math>).</li>
<li>{{mvar|<math>Y}}</math> is a [[topological vector space]] (not necessarily Hausdorff or locally convex) and {.</li>
<li><math>\mathcal{N}</math|1=𝒩}}> is a basis of neighborhoods of 0 in {{mvar|<math>Y}}.</math></li>
<li><math>F</math> is a vector subspace of <math>Y^T = \prod_{t \in T} Y,</math><ref group=note>Because <math>T</math> is just a set that is not yet assumed to be endowed with any vector space structure, <math>F \subseteq Y^T</math> should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined.</ref> which denotes the set of all <math>Y</math>-valued functions <math>f : T \to Y</math> with ___domain <math>T.</math></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>
===𝒢-topology===
:'''Definition and notation''': For any subsets {{mvar|G}} of {{mvar|X}} and {{mvar|N}} of {{mvar|Y}}, let
:"{{math|1=𝒰(''G'', ''N'') := { ''f'' ∈ ''F'' : ''f'' (''G'') ⊆ ''N'' }}}.
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
=== Basic neighborhoods at the origin ===
For any subsets <math>G \subseteq T</math> and <math>N \subseteq Y,</math> let
<math display="block">\mathcal{U}(G, N) := \{f \in F : f(G) \subseteq N\}.</math>
The family
Henceforth assume that {{math|1=''G'' ∈ 𝒢}} and {{math|1=''N'' ∈ 𝒩}}.
<math display="block">\{ \mathcal{U}(G, N) : G \in \mathcal{G}, N \in \mathcal{N} \}</math>
forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set <math>\mathcal{U}(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 <math>F,</math> where this topology is {{em|not}} necessarily a vector topology (that is, it might not make <math>F</math> into a TVS).
This topology does not depend on the neighborhood basis <math>\mathcal{N}</math> that was chosen and it is known as the '''topology of uniform convergence on the sets in <math>\mathcal{G}</math>''' or as the '''<math>\mathcal{G}</math>-topology'''.{{sfn|Schaefer|Wolff|1999|pp=79-88}}
However, this name is frequently changed according to the types of sets that make up <math>\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>\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>\mathcal{G}</math> is the collection of compact subsets of <math>T</math> (and <math>T</math> is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of <math>T.</math></ref>).
A subset <math>\mathcal{G}_1</math> of <math>\mathcal{G}</math> is said to be '''fundamental with respect to <math>\mathcal{G}</math>''' if each <math>G \in \mathcal{G}</math> is a subset of some element in <math>\mathcal{G}_1.</math>
;Properties
In this case, the collection <math>\mathcal{G}</math> can be replaced by <math>\mathcal{G}_1</math> without changing the topology on <math>F.</math>{{sfn|Schaefer|Wolff|1999|pp=79-88}}
One may also replace <math>\mathcal{G}</math> with the collection of all subsets of all finite unions of elements of <math>\mathcal{G}</math> without changing the resulting <math>\mathcal{G}</math>-topology on <math>F.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}
Call a subset <math>B</math> of <math>T</math> '''<math>F</math>-bounded''' if <math>f(B)</math> is a bounded subset of <math>Y</math> for every <math>f \in F.</math>{{sfn|Jarchow|1981|pp=43-55}}
<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>
{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=79-88}}{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
;Algebraic relations
The <math>\mathcal{G}</math>-topology on <math>F</math> is compatible with the vector space structure of <math>F</math> if and only if every <math>G \in \mathcal{G}</math> is <math>F</math>-bounded;
that is, if and only if for every <math>G \in \mathcal{G}</math> and every <math>f \in F,</math> <math>f(G)</math> is [[Bounded set (topological vector space)|bounded]] in <math>Y.</math>
}}
'''Properties'''
Properties of the basic open sets will now be described, so assume that <math>G \in \mathcal{G}</math> and <math>N \in \mathcal{N}.</math>
Then <math>\mathcal{U}(G, N)</math> is an [[Absorbing set|absorbing]] subset of <math>F</math> if and only if for all <math>f \in F,</math> <math>N</math> absorbs <math>f(G)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
If <math>N</math> is [[Balanced set|balanced]]{{sfn|Narici|Beckenstein|2011|pp=371-423}} (respectively, [[Convex set|convex]]) then so is <math>\mathcal{U}(G, N).</math>
The equality
<math>\mathcal{U}(\varnothing, N) = F</math>
always holds.
If <math>s</math> is a scalar then <math>s \mathcal{U}(G, N) = \mathcal{U}(G, s N),</math> so that in particular, <math>- \mathcal{U}(G, N) = \mathcal{U}(G, - N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}
Moreover,{{sfn|Narici|Beckenstein|2011|pp=19-45}}
<math display=block>\mathcal{U}(G, N) - \mathcal{U}(G, N) \subseteq \mathcal{U}(G, N - N)</math>
and similarly{{sfn|Jarchow|1981|pp=43-55}}
<math display=block>\mathcal{U}(G, M) + \mathcal{U}(G, N) \subseteq \mathcal{U}(G, M + N).</math>
For any subsets <math>G, H \subseteq X</math> and any non-empty subsets <math>M, N \subseteq Y,</math>{{sfn|Jarchow|1981|pp=43-55}}
<math display=block>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math>
which implies:
<ul>
<li>Forif any<math>M scalar\subseteq {{mvar|s}}, {{N</math|1=''s''𝒰(''G'',> ''N'')then = 𝒰<math>\mathcal{U}(''G'', ''sN''M)}}; so in particular,\subseteq \mathcal{{math|1=-𝒰U}(''G'', ''N'') = 𝒰(''G'', -''N'')}}.</math>{{sfn | Narici | Beckenstein|2011 | pp=371-423}}</li>
<li>if <math>G \subseteq H</math> then <math>\mathcal{U}(H, N) \subseteq \mathcal{U}(G, N).</math></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:
<li>For any <math>M, N \in \mathcal{N}</math> and subsets <math>G, H, K</math> of <math>T,</math> if <math>G \cup H \subseteq K</math> then <math>\mathcal{U}(K, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N).</math></li>
<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>
For any family <math>\mathcal{S}</math> of subsets of <math>T</math> and any family <math>\mathcal{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} <math display="block">\mathcal{U}\left(\bigcup_{S \in \mathcal{S}} S, N\right) = \bigcap_{S \in \mathcal{S}} \mathcal{U}(S, N) \qquad \text{ and } \qquad \mathcal{U}\left(G, \bigcap_{M \in \mathcal{M}} M\right) = \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math>
=== {{math|1=𝒢}}-topology ===
===Uniform structure===
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>
{{See also|Uniform space}}
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>).
For any <math>G \subseteq T</math> and <math>U \subseteq Y \times Y</math> be any [[Uniform space|entourage]] of <math>Y</math> (where <math>Y</math> is endowed with its [[Complete topological vector space#Canonical uniformity|canonical uniformity]]), let
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>}}.
<math display=block>\mathcal{W}(G, U) ~:=~ \left\{(u, v) \in Y^T \times Y^T ~:~ (u(g), v(g)) \in U \; \text{ for every } g \in G\right\}.</math>
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}}
Given <math>G \subseteq T,</math> the family of all sets <math>\mathcal{W}(G, U)</math> as <math>U</math> ranges over any fundamental system of entourages of <math>Y</math> forms a fundamental system of entourages for a uniform structure on <math>Y^T</math> called {{em|the uniformity of uniform converges on <math>G</math>}} or simply {{em|the <math>G</math>-convergence uniform structure}}.{{sfn|Grothendieck|1973|pp=1-13}}
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}}
The {{em|<math>\mathcal{G}</math>-convergence uniform structure}} is the least upper bound of all <math>G</math>-convergence uniform structures as <math>G \in \mathcal{G}</math> ranges over <math>\mathcal{G}.</math>{{sfn|Grothendieck|1973|pp=1-13}}
:'''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'' ∈\in ''F''}}</math> and let {{<math|1=''f''<sub>•</sub>f_{\bull} = \left(''f''<sub>''i''</sub>f_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} 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|1=''f''<sub>•f_{\bull}</submath>}} '''converges uniformly to {{mvar|<math>f}}</math> on {{mvar|<math>G}}</math>''' if for every {{<math|1=''>N'' ∈\in 𝒩}\mathcal{N}</math> there exists some {{<math|1=''i''<sub>0i_0 \in I</submath> ∈ ''I''}} such that for every {{<math|1=''>i'' ∈\in ''I''}}</math> satisfying {{<math|1=''>i'' ≥\geq ''i''<sub>0i_0,I</submath>}}, {{<math|1=''f''<sub>''i''</sub>f_i - ''f'' ∈\in 𝒰\mathcal{U}(''G'', ''N'')}}</math> (or equivalently, {{<math|1=''f''<sub>''i''</sub>f_i(''g'') - ''f'' (''g'') ∈\in ''N''}}</math> for every {{<math|1=''>g'' ∈\in ''G''}}</math>). {{sfn|Jarchow|1981|pp=43-55}}
{{Math theorem|name=Theorem{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=
If {{<math|1=''>f'' ∈\in ''F''}}</math> and if {{math|1=''f''<sub>•</submath>f_{\bull} = \left(''f''<sub>''i''</sub>f_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} is a net in {{mvar|<math>F}},</math> then {{<math|1=''>f_{\bull} \to f''<sub>•</submath> → ''f''}} in the {{<math|1=𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> if and only if for every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}, {{</math|1=''f''> <submath>•f_{\bull}</submath>}} converges uniformly to {{mvar|<math>f}}</math> on {{mvar|<math>G}}.</math>
}}
=== Inherited properties ===
;'''Local convexity'''
If {{mvar|<math>Y}}</math> is [[locally convex]] then so is the {{<math|1=𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> and if {{math|1=(''p''<sub>''i''</submath>\left(p_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} is a family of continuous seminorms generating this topology on {{mvar|<math>Y}}</math> then the {{<math|1=𝒢}>\mathcal{G}</math>-topology is induced by the following family of seminorms:
<math display="block">p_{G,i}(f) := \sup_{x \in G} p_i(f(x)),</math>
:{{math|1=''p''<sub>''G'',''i''</sub>( ''f'' ) {{=}}}} {{underset|{{math|1=''x'' ∈ ''G''}}|sup}} {{math|1=''p''<sub>''i''</sub>( ''f''(''x''))}},
as {{mvar|<math>G}}</math> varies over <math>\mathcal{{G}</math|1=𝒢}}> and {{mvar|<math>i}}</math> varies over {{mvar|<math>I}}</math>.{{sfn | Schaefer | Wolff|1999 | p=81}}
;'''Hausdorffness'''
If {{mvar|<math>Y}}</math> is [[Hausdorff space|Hausdorff]] and {{<math|1=''>T'' = \bigcup_{{underset|''G'' ∈\in 𝒢|\mathcal{{big|∪}}G}} ''G''}}</math> then the <math>\mathcal{{G}</math|1=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|1=''>Y''<sup>''^T''</supmath>}} consisting of all continuous maps that are bounded on every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}</math> and if {{<math|1=>\bigcup_{{underset|''G'' ∈\in 𝒢|\mathcal{{big|∪}}G}} ''G''}}</math> is dense in {{mvar|<math>T}}</math> then the {{<math|1=𝒢}>\mathcal{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|1=𝒢}}>-topology if and only if for every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G},</math> {{<math|1=''>H''(''G'') := \bigcup_{{underset|''h'' ∈\in ''H''|{{big|∪}}}} ''h''(''G'')}}</math> 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|1=𝒢}}> be the set of all finite subsets of {{mvar|<math>T}}</math> then the <math>\mathcal{{G}</math|1=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|1=''Y''<supmath>''Y^T''</supmath>}} when {{math|1=''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|1=>C(''X'')}}</math> is the space of all real (or complex) valued continuous functions on {{mvar|<math>X}},</math> the topology of pointwise convergence on {{<math|1=>C(''X'')}}</math> 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|1=𝒢}}> will be a non-empty collection of subsets of {{mvar|<math>X}}</math> [[Directed set|directed]] by inclusion.
<math>L(X; Y)</math> will denote the vector space of all continuous linear maps from <math>X</math> into <math>Y.</math> If <math>L(X; Y)</math> is given the <math>\mathcal{G}</math>-topology inherited from <math>Y^X</math> then this space with this topology is denoted by <math>L_{\mathcal{G}}(X; Y)</math>.
The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space <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; \mathbb{F})</math> and is denoted by <math>X^{\prime}</math>.
The <math>\mathcal{G}</math>-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 \mathcal{G}</math> and all <math>f \in L(X; Y)</math> the set <math>f(G)</math> is bounded in <math>Y,</math> which we will assume to be the case for the rest of the article.
:'''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'')}}.
Note in particular that this is the case if <math>\mathcal{G}</math> consists of [[Bounded set (topological vector space)|(von-Neumann) bounded]] subsets of <math>X.</math>
===Assumptions on 𝒢===
:'''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|{{'}}}}}}.
'''Assumptions that guarantee a vector topology'''
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}}.
* (<math>\mathcal{G}</math> is directed): <math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by (subset) inclusion. That is, for any <math>G, H \in \mathcal{G},</math> there exists <math>K \in \mathcal{G}</math> such that <math>G \cup H \subseteq K</math>.
=== Assumptions on 𝒢 ===
The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
;Assumptions that guarantee a vector topology
The next assumption will guarantee that the sets <math>\mathcal{U}(G, N)</math> are [[Balanced set|balanced]].
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
* (<math>N \in \mathcal{N}</math> are balanced): <math>\mathcal{N}</math> is a neighborhoods basis of the origin in <math>Y</math> that consists entirely of [[Balanced set|balanced]] sets.
:'''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 following assumption is very commonly made because it will guarantee that each set <math>\mathcal{U}(G, N)</math> is absorbing in <math>L(X; Y).</math>
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.
* (<math>G \in \mathcal{G}</math> are bounded): <math>\mathcal{G}</math> is assumed to consist entirely of bounded subsets of <math>X.</math>
:'''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 next theorem gives ways in which <math>\mathcal{G}</math> can be modified without changing the resulting <math>\mathcal{G}</math>-topology on <math>Y.</math>
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'')}}.
{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=371-423}}|math_statement=
:'''Assumption''' ({{math|1=''G'' ∈ 𝒢}} are bounded): {{math|1=𝒢}} is assumed to consist entirely of bounded subsets of {{mvar|X}}.
Let <math>\mathcal{G}</math> be a non-empty collection of bounded subsets of <math>X.</math> Then the <math>\mathcal{G}</math>-topology on <math>L(X; Y)</math> is not altered if <math>\mathcal{G}</math> is replaced by any of the following collections of (also bounded) subsets of <math>X</math>:
;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>\mathcal{{G}</math|1=𝒢}}>;</li>
<li>all scalar multiples of all sets in <math>\mathcal{{G}</math|1=𝒢}}>;</li>
<li>all finite [[Minkowski sum]]s of sets in <math>\mathcal{{G}</math|1=𝒢}}>;</li>
<li>the [[Balanced set|balanced hull]] of every set in <math>\mathcal{{G}</math|1=𝒢}}>;</li>
<li>the closure of every set in <math>\mathcal{{G}</math|1=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|1=𝒢}G}.</math></li>
</ol>
}}
;'''Common assumptions'''
Some authors (e.g. Narici) require that <math>\mathcal{{G}</math|1=𝒢}}> satisfy the following condition, which implies, in particular, that <math>\mathcal{{G}</math|1=𝒢}}> is [[Directed set|directed]] by subset inclusion:
:<math>\mathcal{{G}</math|1=𝒢}}> is assumed to be closed with respect to the formation of subsets of finite unions of sets in <math>\mathcal{{G}</math|1=𝒢}}> (i.e. every subset of every finite union of sets in <math>\mathcal{{G}</math|1=𝒢}}> belongs to <math>\mathcal{{G}</math|1=𝒢}}>).
Some authors (e.g. Trèves) require that {{mathsfn|1Trèves|2006|loc=𝒢Chapter 32}}) require that <math>\mathcal{G}</math> be directed under subset inclusion and that it satisfy the following condition:
:If {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{mvar|<math>s}}</math> is a scalar then there exists a {{<math|1=''>H'' ∈\in 𝒢}\mathcal{G}</math> such that {{<math|1=''sG''>s ⊆G \subseteq ''H''}}.</math>
If <math>\mathcal{{G}</math|1=𝒢}}> is a [[bornology]] on {{mvar|<math>X}},</math> which is often the case, then these axioms are satisfied.
If <math>\mathcal{{G}</math|1=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'''
A subset of a TVS <math>X</math> whose [[linear span]] is a [[dense set|dense subset]] of <math>X</math> is said to be a [[Total set|total subset]] of <math>X.</math>
:'''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 <math>\mathcal{G}</math> is a family of subsets of a TVS <math>T</math> then <math>\mathcal{G}</math> is said to be '''[[Total set|total in <math>T</math>]]''' if the [[linear span]] of <math>\bigcup_{G \in \mathcal{G}} G</math> is dense in <math>T.</math>{{sfn|Schaefer|Wolff|1999|p=80}}
If {{mvar|<math>F}}</math> is the vector subspace of {{<math|1=''>Y''<sup>''^T''</supmath>}} consisting of all continuous linear maps that are bounded on every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G},</math> then the {{<math|1=𝒢}>\mathcal{G}</math>-topology on {{mvar|<math>F}}</math> is Hausdorff if {{mvar|<math>Y}}</math> is Hausdorff and {{<math|1=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|1=𝒢}>\mathcal{G}</math> 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|1=''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{mvar|<math>s}}</math> is a scalar then there exists a {{<math|1=''>H'' ∈\in 𝒢}\mathcal{G}</math> such that {{<math|1=''sG''>s ⊆G \subseteq ''H''}}. </math>
<ul>
<li>{{math|1=L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> is [[Complete topological vector space|complete]] if
{{ordered list|
| {{mvar|<math>X}}</math> is locally convex and Hausdorff,
| {{mvar|<math>Y}}</math> is complete, and
| whenever {{<math|1=''>u'' : ''X'' →\to ''Y''}}</math> is a linear map then {{mvar|<math>u}}</math> restricted to every set {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}</math> is continuous implies that {{mvar|<math>u}}</math> is continuous,
}}</li>
<li>If {{mvar|<math>X}}</math> is a Mackey space then {{math|1=L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> 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|1=L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> 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 [[barreledBarreled 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|1=𝒢}>\mathcal{G}</math> covers {{mvar|<math>X}}</math> then every closed [[Equicontinuous linear maps|equicontinuous subset]] of {{<math|1=>L(''X''; ''Y'')}}</math> is complete in {{math|1=L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> and {{<math|1=L<sub>𝒢</sub>L_{\mathcal{G}}(''X''; ''Y'')}}</math> 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|1=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|1=''>G'' ∈\in 𝒢}\mathcal{G}.</math> If {{mvar|<math>Y}}</math> is [[quasi-complete]] (resp.respectively, [[Complete topological vector space|complete]]) then so is {{math|1=L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math>.{{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|1=>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|1=L<sub>𝒢</sub>L_{\mathcal{G}}(''X''; ''Y'')}}</math>;</li>
<li>For every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G},</math> {{<math|1=''>H''(''G'') := \bigcup_{{underset|''h'' ∈\in ''H''|{{big|∪}}}} ''h''(''G'')}}</math> is bounded in {{mvar|<math>Y}}</math>;{{sfn | Schaefer | Wolff|1999 | p=81}}</li>
<li>For every neighborhood {{mvar|<math>V}}</math> of 0the origin in {{mvar|<math>Y}}</math> the set {{<math|1=>\bigcap_{{underset|''h'' ∈\in ''H''|{{big|∩}}}} ''h'' <sup>^{-1}(V)</supmath>(''V'')}} [[Absorbing set|absorbs]] every {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}.</math></li>
</ol>
If <math>\mathcal{G}</math> is a collection of bounded subsets of <math>X</math> whose union is [[Total set|total]] in <math>X</math> then every [[Equicontinuous linear maps|equicontinuous subset]] of <math>L(X; Y)</math> is bounded in the <math>\mathcal{G}</math>-topology.{{sfn|Schaefer|Wolff|1999|p=83}}
Furthermore,
Furthermore, if <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then
<ul>
<li>If {{mvar|X}} and {{mvar|Y}} are locally convex Hausdorff space and if {{mvar|<math>H}}</math> is bounded in {{<math|1=L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math> (i.e.that is, 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|X}} and {{mvar|Y}} are locally convex Hausdorff spaces and if {{mvar|<math>X}}</math> is [[Quasi-complete space|quasi-complete]] (i.e.meaning that closed and bounded subsets are complete), then the bounded subsets of {{<math|1=>L(''X''; ''Y'')}}</math> are identical for all <math>\mathcal{{G}</math|1=𝒢}}>-topologies where <math>\mathcal{{G}</math|1=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></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>
=== Examples ===
{| class="wikitable"
|-
! <math>\mathcal{{math|1=𝒢G} ⊆\subseteq 𝒫\wp(''X'')}}</math> ("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
! Alternative name
|-
| finite subsets of {{mvar|<math>X}}</math>
| {{<math|1=L<sub>σ</sub>L_{\sigma}(''X''; ''Y'')}}</math>
| 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|1=L<sub>γ</sub>L_{\gamma}(''X''; ''Y'')}}</math>
| compact convex convergence
|
|-
| compact subsets of {{mvar|<math>X}}</math>
| {{math|1=L<sub>c</submath>L_c(''X''; ''Y'')}}</math>
| compact convergence
|
|-
| bounded subsets of {{mvar|<math>X}}</math>
| {{math|1=L<sub>b</submath>L_b(''X''; ''Y'')}}</math>
| bounded convergence
| strong topology
|}
==== The topology of pointwise convergence {{math|1=L<sub>σ</sub>(''X''; ''Y'')}} ====
By letting {{<math|1=𝒢}>\mathcal{G}</math> be the set of all finite subsets of {{mvar|<math>X}},</math> {{<math|1=>L(''X''; ''Y'')}}</math> will have the '''weak topology on {{<math|1=>L(''X''; ''Y'')}}</math>''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and {{<math|1=>L(''X''; ''Y'')}}</math> with this topology is denoted by {{<math|1=L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math>.
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|1=>L(''X''; ''Y'')}}</math> is called '''simply bounded''' or '''weakly bounded''' if it is bounded in {{<math|1=L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math>.
The weak-topology on {{<math|1=>L(''X''; ''Y'')}}</math> has the following properties:
<ul>
<li>If {{mvar|<math>X}}</math> is [[Separable space|separable]] (i.e.that is, it has a countable dense subset) and if {{mvar|<math>Y}}</math> is a metrizable topological vector space then every equicontinuous[[Equicontinuous subsetlinear {{mvarmaps|H}}equicontinuous ofsubset]] {{math|1=L<submath>𝜎H</submath> of <math>L_{\sigma}(''X''; ''Y'')}}</math> 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|1=>L(''X''; ''Y'')}},</math> the topology of pointwise convergence is metrizable.</li>
<li>Let {{math|1=''Y''<supmath>''Y^X''</supmath>}} denote the space of all functions from {{mvar|<math>X}}</math> into {{mvar|<math>Y}}.</math> If {{<math|1=>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|1=''Y''<supmath>''Y^X''</supmath>}}.
* In addition, {{<math|1=>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|1=>L(''X''; ''Y'')}}</math> is bounded when {{<math|1=>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|1=>L(''X''; ''Y'')}}</math> are identical for all <math>\mathcal{{G}</math|1=𝒢}}>-topologies on {{<math|1=>L(''X''; ''Y'')}}</math> such that <math>\mathcal{{G}</math|1=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 linear maps|equicontinuous subset]] of {{<math|1=>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|1=>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|1=>L(''X''; ''Y'')}}</math> is equicontinuous.{{sfn | Schaefer | Wolff|1999 | p=83}}</li>
<li>On an equicontinuous subset {{mvar|<math>H}}</math> of {{<math|1=>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|1=L<sub>c</sub>(''X''; ''Y'')}} ====
By letting {{<math|1=𝒢}>\mathcal{G}</math> be the set of all compact subsets of {{mvar|<math>X}},</math> {{<math|1=>L(''X''; ''Y'')}}</math> will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and {{<math|1=>L(''X''; ''Y'')}}</math> with this topology is denoted by {{<math|1=L<sub>c</sub>L_c(''X''; ''Y'')}}</math>.
The topology of compact convergence on {{<math|1=>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 metrictopological space#Topologically completevector spacesspace|complete]] locally convex Hausdorff space then {{<math|1=L<sub>c</sub>L_c(''X''; ''Y'')}}</math> is complete.</li>
<li>On [[Equicontinuous linear maps|equicontinuous subsets]] of {{<math|1=>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|1=L<sub>c</sub>L_c(''X''; ''Y'')}}</math> and {{<math|1=L<sub>b</sub>L_b(''X''; ''Y'')}}</math> have identical topologies.</li>
</ul>
==== Topology of bounded convergence {{math|1=L<sub>b</sub>(''X''; ''Y'')}} ====
By letting {{<math|1=𝒢}>\mathcal{G}</math> be the set of all bounded subsets of {{mvar|<math>X}},</math> {{<math|1=>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|1=>L(''X''; ''Y'')}}</math> with this topology is denoted by {{<math|1=L<sub>b</sub>L_b(''X''; ''Y'')}}</math>.{{sfn | Narici | Beckenstein|2011 | pp=371-423}}
The topology of bounded convergence on {{<math|1=>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 metrictopological space#Topologicallyvector complete spacesspace|complete]] locally convex Hausdorff space then {{<math|1=L<sub>b</sub>L_b(''X''; ''Y'')}}</math> is complete.</li>
<li>If {{mvar|<math>X}}</math> and {{mvar|<math>Y}}</math> are both normed spaces then the topology on {{<math|1=>L(''X''; ''Y'')}}</math> induced by the usual operator norm is identical to the topology on {{<math|1=L<sub>b</sub>L_b(''X''; ''Y'')}}</math>.{{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|1=''>X'' ^{{big|{{'}}}}}\prime}</math> is identical to the topology of bounded convergence on {{<math|1=''>X'' ^{{big|{{'}}}}}\prime}</math>.</li>
<li>Every equicontinuous subset of {{<math|1=>L(''X''; ''Y'')}}</math> is bounded in {{math|1=L<sub>b</submath>L_b(''X''; ''Y'')}}</math>.</li>
</ul>
== Polar topologies ==
{{Main|Polar topology}}
Throughout, we assume that {{mvar|<math>X}}</math> is a TVS.
=== {{math|1=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 ''{{em|weakly''}} bounded subsets (e.g. if {{mvar|<math>X}}</math> is a Hausdorff locally convex space), then a <math>\mathcal{{G}</math|1=𝒢}}>-topology on {{<math|1=''>X''^{{big|{{'}}}}}\prime}</math> (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a <math>\mathcal{{G}</math|1=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 ''{{em|not''}} exactly the same as its ''{{em|weakly''}} bounded subsets, then the notion of "bounded in {{mvar|<math>X}}</math>" is stronger than the notion of "{{<math|1=σ>\sigma\left(''X'', ''X''^{{big|{{'}}}\prime}\right)}}</math>-bounded in {{mvar|<math>X}}</math>" (i.e. bounded in {{mvar|<math>X}}</math> implies {{<math|1=σ>\sigma\left(''X'', ''X''^{{big|{{'}}}\prime}\right)}}</math>-bounded in {{mvar|<math>X}}</math>) so that a <math>\mathcal{{G}</math|1=𝒢}}>-topology on {{<math|1=''>X''^{{big|{{'}}}}}\prime}</math> (as defined in this article) is ''{{em|not''}} necessarily a polar topology.
One important difference is that polar topologies are always locally convex while <math>\mathcal{{G}</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]].
We list here some of the most common polar topologies.
=== 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|1=𝛥>\Delta(''Y'', ''X'')}}</math> denotes a polar topology on {{mvar|<math>Y}}</math> then {{mvar|<math>Y}}</math> endowed with this topology will be denoted by {{<math|1=''Y''<sub>𝛥Y_{\Delta(''Y'', ''X'')}</submath>}} or simply {{<math|1=''Y''<sub>𝛥Y_{\Delta}</submath>}} (e.g. for {{<math|1=σ>\sigma(''Y'', ''X'')}}</math> we'd would have {{<math|1=𝛥>\Delta {{=}} σ}}\sigma</math> so that {{<math|1=''Y''<sub>σY_{\sigma(''Y'', ''X'')}</submath>}} and {{<math|1=''Y''<sub>σY_{\sigma}</submath>}} all denote {{mvar|<math>Y}}</math> with endowed with {{<math|1=σ>\sigma(''Y'', ''X'')}}</math>).
{| class="wikitable"
|-
! ><math>\mathcal{{math|1=𝒢G} ⊆\subseteq 𝒫\wp(''X'')}}</math><br/>("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
! Alternative name
|-
| finite subsets of {{mvar|<math>X}}</math>
| {{<math|1=σ>\sigma(''Y'', ''X'')}}</math><br/>{{<math|1=>s(''Y'', ''X'')}}</math>
| pointwise/simple convergence
| [[Weak topology|weak/weak* topology]]
|-
| {{<math|1=σ>\sigma(''X'', ''Y'')}}</math>-compact [[Absolutely convex set|disk]]s
| {{<math|1=τ>\tau(''Y'', ''X'')}}</math>
|
| [[Mackey topology]]
|-
| {{<math|1=σ>\sigma(''X'', ''Y'')}}</math>-compact convex subsets
| {{<math|1=γ>\gamma(''Y'', ''X'')}}</math>
| compact convex convergence
|
|-
| {{<math|1=σ>\sigma(''X'', ''Y'')}}</math>-compact subsets<br/>(or balanced {{<math|1=σ>\sigma(''X'', ''Y'')}}</math>-compact subsets)
| {{<math|1=>c(''Y'', ''X'')}}</math>
| compact convergence
|
|-
| {{<math|1=σ>\sigma(''X'', ''Y'')}}</math>-bounded subsets
| {{<math|1=>b(''Y'', ''X'')}}</math><br/>{{<math|1=𝛽>\beta(''Y'', ''X'')}}</math>
| bounded convergence
| [[Strong dual space|strong topology]]
|}
== {{math|1=𝒢-ℋ}}- topologies on spaces of bilinear maps ==
We will let {{<math|1=ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> denote the space of separately continuous bilinear maps and {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math> 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|1=>L(''X''; ''Y'')}}</math> we can place a topology on <math>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'')}}</math> and {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math>.
Let <math>\mathcal{{G}</math|1=𝒢}}> (resp.respectively, <math>\mathcal{{H}</math|1=ℋ}}>) be a family of subsets of {{mvar|<math>X}}</math> (resp.respectively, {{mvar|<math>Y}}</math>) containing at least one non-empty set.
Let {{<math|1=𝒢>\mathcal{G} ×\times ℋ}\mathcal{H}</math> denote the collection of all sets {{<math|1=''>G'' ×\times ''H''}}</math> where {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G},</math> {{<math|1=''>H'' ∈\in ℋ}\mathcal{H}.</math>
We can place on {{math|1=''Z''<supmath>''Z^{X'' ×\times ''Y''}</supmath>}} the {{<math|1=𝒢>\mathcal{G} ×\times ℋ}\mathcal{H}</math>-topology, and consequently on any of its subsets, in particular on {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math> and on {{<math|1=ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math>.
This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math|1=𝒢-ℋ}}>-topology''' or as the '''topology of uniform convergence on the products {{<math|1=''>G'' ×\times ''H''}}</math> of <math>\mathcal{{math|1=𝒢G} ×\times ℋ}\mathcal{H}</math>'''.
However, as before, this topology is not necessarily compatible with the vector space structure of <math>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'')}}</math> or of {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math> without the additional requirement that for all bilinear maps, {{mvar|<math>b}}</math> in this space (that is, in {{<math|1=ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> or in {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math>) and for all {{<math|1=''>G'' ∈\in 𝒢}\mathcal{G}</math> and {{<math|1=''>H'' ∈\in ℋ}\mathcal{H},</math> the set {{<math|1=>b(''G'', ''H'')}}</math> is bounded in {{mvar|<math>X}}.</math>
If both <math>\mathcal{{G}</math|1=𝒢}}> and <math>\mathcal{{H}</math|1=ℋ}}> consist of bounded sets then this requirement is automatically satisfied if we are topologizing {{<math|1=>B(''X'', ''Y''; ''Z'')}}</math> but this may not be the case if we are trying to topologize <math>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'')}}</math>.
The {{<math|1=𝒢>\mathcal{G}-ℋ}\mathcal{H}</math>-topology on {{<math|1=ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> will be compatible with the vector space structure of <math>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'')}}</math> if both <math>\mathcal{{G}</math|1=𝒢}}> and <math>\mathcal{{H}</math|1=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>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'') = B(''X'', ''Y''; ''Z'')}}.</math>
* {{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|1=𝒢>\mathcal{G}^{'}}}\prime}</math> and {{<math|1=ℋ >\mathcal{H}^{'}}}\prime}</math> be the collections of [[Equicontinuous linear functionals|equicontinuous subsets]] of {{<math|1=''>X''{{big|{^{'}}}}}\prime}</math> and {{<math|1=''Y''>X^{{big|{{'}}}}}\prime}</math>, respectively.
Then the {{<math|1=𝒢>\mathcal{G}^{'}\prime}-ℋ \mathcal{H}^{'}}}\prime}</math>-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|1=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 (orrespectively, both Banach) then so is <math>\mathcal{B}_{\epsilon}\left( X^{\prime}_{\sigma}, Y^{\prime}_{\sigma} \right)</math>
== See also ==
* {{annotated link|Bornological space}}
* {{annotated link|Dual system}}
* {{annotated link|Dual topology}}
* {{annotated link|LocallyList convexof topological vector spacetopologies}}
* {{annotated link|Modes of convergence}}
* {{annotated link|Operator norm}}
* {{annotated link|Polar topology}}
* {{annotated link|Strong dual space}}
* {{annotated link|Strong topology (polar topology)}}
* {{annotated link|Topological vector space}}
* {{annotated link|Topologies on the set of operators on a Hilbert space}}
* {{annotated link|Uniform convergence}}
* {{annotated link|Uniform space}}
* {{annotated link|Weak topology}}
** {{annotated link|Vague topology}}
==References==
{{reflist|group=note}}
{{reflist|group=proof}}
{{reflist}}
== References Bibliography==
{{Reflist}}
* {{Narici BeckensteinGrothendieck Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici Grothendieck| 2011 1973| p=}} -->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn|Hogbe-Nlend|1977|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}}
* {{SchaeferJarchow WolffLocally Topological VectorConvex Spaces|edition=2}} <!-- {{sfn | Schaefer Jarchow| 1999 1981| p=}} -->
* {{TrèvesKhaleelulla FrançoisCounterexamples in Topological vectorVector spaces, distributions and kernelsSpaces}} <!-- {{sfn | Trèves Khaleelulla| 2006 1982| p=}} -->
* {{KhaleelullaNarici Counterexamples inBeckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Khaleelulla Narici| {{{yearBeckenstein| 1982 }}} 2011| p=}} -->
* {{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 Analysisanalysis}}
{{Duality and spaces of linear maps}}
{{DualityInLCTVSs}}
{{Topological vector spaces}}
[[Category:Functional analysis]]
[[Category:Topological vector spaces]]
[[Category:Topology of function spaces]]
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