Topologies on spaces of linear maps: Difference between revisions

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:'''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'' }}}.

=== Basic neighborhoods at the origin ===

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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|𝒩}} that was chosen and it is known as the '''topology of uniform convergence on the sets in {{math|𝒢}}''' or as the '''{{math|𝒢}}-topology'''.{{sfn | Schaefer|Wolff| 1999 | pp=79-88}}

However, this name is frequently changed according to the types of sets that make up {{math|𝒢}} (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details<ref>In practice, {{math|𝒢}} usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, {{math|𝒢}} 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|𝒢₁}} 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|𝒢₁}}.

In this case, the collection {{math|𝒢}} can be replaced by {{math|𝒢₁}} without changing the topology on {{mvar|F}}.{{sfn | Schaefer|Wolff| 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''}}.

{{Math theorem|name=Theorem{{sfn | Schaefer|Wolff| 1999 | pp=79-88}}{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=

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==== Nets and uniform convergence ====

:'''Definition''':{{sfn | Jarchow | 1981 | pp=43-55}} Let {{math|''f'' ∈ ''F''}} and let {{math|1=''f''_• = (''f''_''i'')_{''i'' ∈ ''I''}}} be a [[Net (mathematics)|net]] in {{mvar|F}}. Then for any subset {{mvar|G}} of {{mvar|T}}, say that {{math|''f''_•}} '''converges uniformly to {{mvar|f}} on {{mvar|G}}''' if for every {{math|''N'' ∈ 𝒩}} there exists some {{math|''i''₀ ∈ ''I''}} such that for every {{math|''i'' ∈ ''I''}} satisfying {{math|''i'' ≥ ''i''₀}}, {{math|''f''_''i'' - ''f'' ∈ 𝒰(''G'', ''N'')}} (or equivalently, {{math|''f''_''i''(''g'') - ''f'' (''g'') ∈ ''N''}} for every {{math|''g'' ∈ ''G''}}).

{{Math theorem|name=Theorem{{sfn | Jarchow | 1981 | pp=43-55}}|math_statement=

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If {{mvar|Y}} is [[locally convex]] then so is the {{math|𝒢}}-topology on {{mvar|F}} and if {{math|(''p''_''i'')_{''i'' ∈ ''I''}}} is a family of continuous seminorms generating this topology on {{mvar|Y}} then the {{math|𝒢}}-topology is induced by the following family of seminorms:

:{{math|''p''_''G'',''i''(&thinsp;''f''&thinsp;) {{=}}}} {{underset|{{math|''x'' ∈ ''G''}}|sup}} {{math|''p''_''i''(&thinsp;''f''(''x''))}},

as {{mvar|G}} varies over {{math|𝒢}} and {{mvar|i}} varies over {{mvar|I}}.{{sfn | Schaefer|Wolff| 1999 | p=81}}

;Hausdorffness

If {{mvar|Y}} is [[Hausdorff space|Hausdorff]] and {{math|1=''T'' = {{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} then the {{math|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|''Y''^''T''}} consisting of all continuous maps that are bounded on every {{math|''G'' ∈ 𝒢}} and if {{math|{{underset|''G'' ∈ 𝒢|{{big|∪}}}} ''G''}} is dense in {{mvar|T}} then the {{math|𝒢}}-topology on {{mvar|F}} is Hausdorff.

;Boundedness

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If we let {{math|𝒢}} be the set of all finite subsets of {{mvar|T}} then the {{math|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|''Y''^''T''}} when {{math|''Y''^''T''}} is endowed with the usual [[product topology]].

If {{mvar|X}} is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and {{math|C(''X'')}} is the space of all real (or complex) valued continuous functions on {{mvar|X}}, the topology of pointwise convergence on {{math|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|𝒢}} will be a non-empty collection of subsets of {{mvar|X}} [[Directed set|directed]] by inclusion.

:'''Notation''': {{math|L(''X''; ''Y'')}} will denote the vector space of all continuous linear maps from {{mvar|X}} into {{mvar|Y}}. If {{math|L(''X''; ''Y'')}} is given the {{math|𝒢}}-topology inherited from {{math|''Y''^''X''}} then this space with this topology is denoted by {{math|L_𝒢(''X'', ''Y'')}}.

:'''Notation''': The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space {{mvar|X}} over the field {{math|𝔽}} (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space {{math|L(''X''; 𝔽)}} and is denoted by {{math|''X''{{big|{{'}}}}}}.

The {{math|𝒢}}-topology on {{math|L(''X''; ''Y'')}} is compatible with the vector space structure of {{math|L(''X''; ''Y'')}} if and only if for all {{math|''G'' ∈ 𝒢}} and all {{math|''f'' ∈ L(''X''; ''Y'')}} the set {{math|''f''(''G'')}} is bounded in {{mvar|Y}}, which we will assume to be the case for the rest of the article.

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;Assumptions that guarantee a vector topology

:'''Assumption''' ({{math|𝒢}} is directed): {{math|𝒢}} will be a non-empty collection of subsets of {{mvar|X}} [[Directed set|directed]] by (subset) inclusion. That is, for any {{math|''G'', ''H'' ∈ 𝒢}}, there exists {{math|''K'' ∈ 𝒢}} such that {{math|''G'' ∪ ''H'' ⊆ ''K''}}.

The above assumption guarantees that the collection of sets {{math|𝒰(''G'', ''N'')}} forms a [[filter base]].

The next assumption will guarantee that the sets {{math|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|''N'' ∈ 𝒩}} are balanced): {{math|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|𝒰(''G'', ''N'')}} is absorbing in {{math|L(''X''; ''Y'')}}.

:'''Assumption''' ({{math|''G'' ∈ 𝒢}} are bounded): {{math|𝒢}} is assumed to consist entirely of bounded subsets of {{mvar|X}}.

;Other possible assumptions

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=

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;Hausdorffness

:'''Definition''':{{sfn | Schaefer|Wolff| 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''^''T''}} 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

For the following theorems, suppose that {{mvar|X}} is a topological vector space and {{mvar|Y}} is a [[locally convex]] Hausdorff spaces and {{math|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|''G'' ∈ 𝒢}} and {{mvar|s}} is a scalar then there exists a {{math|''H'' ∈ 𝒢}} such that {{math|''sG'' ⊆ ''H''}}.

<ul>

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<li>{{mvar|H}} is [[Bounded set (topological vector space)|bounded]] in {{math|L_𝒢(''X''; ''Y'')}};</li>

<li>For every {{math|''G'' ∈ 𝒢}}, {{math|1=''H''(''G'') := {{underset|''h'' ∈ ''H''|{{big|∪}}}} ''h''(''G'')}} is bounded in {{mvar|Y}};{{sfn | Schaefer|Wolff| 1999 | p=81}}</li>

<li>For every neighborhood {{mvar|V}} of 0 in {{mvar|Y}} the set {{math|{{underset|''h'' ∈ ''H''|{{big|∩}}}} ''h''&thinsp;^-1−1(''V'')}} [[Absorbing set|absorbs]] every {{math|''G'' ∈ 𝒢}}.</li>

</ol>

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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_𝜎(''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_𝜎(''X''; ''Y'')}}.

The weak-topology on {{math|L(''X''; ''Y'')}} has the following properties:

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==== Topology of bounded convergence {{math|L_b(''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_b(''X''; ''Y'')}}.{{sfn | Narici | Beckenstein | 2011 | pp=371-423}}

The topology of bounded convergence on {{math|L(''X''; ''Y'')}} has the following properties:

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{{Main|Polar topology}}

Throughout, we assume that {{mvar|X}} is a TVS.

=== {{math|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|𝒢}}-topology on {{math|''X''{{big|{{'}}}}}} (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a {{math|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|σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|X}}" (i.e. bounded in {{mvar|X}} implies {{math|σ(''X'', ''X''{{big|{{'}}}})}}-bounded in {{mvar|X}}) so that a {{math|𝒢}}-topology on {{math|''X''{{big|{{'}}}}}} (as defined in this article) is ''not'' necessarily a polar topology.

One important difference is that polar topologies are always locally convex while {{math|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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=== List of polar topologies ===

Suppose that {{mvar|X}} is a TVS whose bounded subsets are the same as its weakly bounded subsets.

:'''Notation''': If {{math|𝛥(''Y'', ''X'')}} denotes a polar topology on {{mvar|Y}} then {{mvar|Y}} endowed with this topology will be denoted by {{math|''Y''_{𝛥(''Y'', ''X'')}}} or simply {{math|''Y''_𝛥}} (e.g. for {{math|σ(''Y'', ''X'')}} we'd have {{math|𝛥 {{=}} σ}} so that {{math|''Y''_{σ(''Y'', ''X'')}}} and {{math|''Y''_σ}} all denote {{mvar|Y}} with endowed with {{math|σ(''Y'', ''X'')}}).

{| class="wikitable"