Topologies on spaces of linear maps: Difference between revisions

Revision as of 04:07, 11 September 2020

Revision as of 15:50, 17 October 2020

Line 49:

Then the set {{math|{𝒰(''G'', ''N'') : ''G'' ∈ 𝒢, ''N'' ∈ 𝒩}}} forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set {{math|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|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=

The {{math|𝒢}}-topology on {{mvar|F}} is compatible with the vector space structure of {{mvar|F}} if and only if every {{math|''G'' ∈ 𝒢}} is {{mvar|F}}-bounded;

that is, if and only if for every {{math|''G'' ∈ 𝒢}} and every {{math|''f'' ∈ ''F''}}, {{math|''f'' (''G'')}} is [[Bounded set (topological vector space)|bounded]] in {{mvar|Y}}.

Line 77:

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

Line 88:

;Boundedness

A subset {{mvar|H}} of {{mvar|F}} is [[Bounded set (topological vector space)|bounded]] in the {{math|𝒢}}-topology if and only if 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}}

=== Examples of 𝒢-topologies ===

Line 161:

;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}}

Line 178:

<li>If {{mvar|X}} is a Mackey space then {{math|L_𝒢(''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|L_𝒢(''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|𝒢}} covers {{mvar|X}} then every closed equicontinuous subset of {{math|L(''X''; ''Y'')}} is complete in {{math|L_𝒢(''X''; ''Y'')}} and {{math|L_𝒢(''X''; ''Y'')}} is quasi-complete.{{sfn | Schaefer |Wolff| 1999 | p=83}}</li>

<li>Let {{mvar|X}} be a [[bornological space]], {{mvar|Y}} a locally convex space, and {{math|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|''G'' ∈ 𝒢}}. If {{mvar|Y}} is [[quasi-complete]] (resp. complete) then so is {{math|L_𝒢(''X''; ''Y'')}}.{{sfn | Schaefer |Wolff| 1999 | p=117}}</li>

</ul>

Line 185:

Let {{mvar|X}} and {{mvar|Y}} be topological vector spaces and {{mvar|H}} be a subset of {{math|L(''X''; ''Y'')}}.

Then the following are equivalent:{{sfn | Schaefer |Wolff| 1999 | p=81}}

<ol>

<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(''V'')}} [[Absorbing set|absorbs]] every {{math|''G'' ∈ 𝒢}}.</li>

</ol>

Line 194:

Furthermore,

<ul>

<li>If {{mvar|X}} and {{mvar|Y}} are locally convex Hausdorff space and if {{mvar|H}} is bounded in {{math|L_𝜎(''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 |Wolff| 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|L(''X''; ''Y'')}} are identical for all {{math|𝒢}}-topologies where {{math|𝒢}} is any family of bounded subsets of {{mvar|X}} covering {{mvar|X}}.{{sfn | Schaefer |Wolff| 1999 | p=82}}</li>

<li>If {{math|𝒢}} is any collection of bounded subsets of {{mvar|X}} whose union is total in {{mvar|X}} then every equicontinuous subset of {{math|L(''X''; ''Y'')}} is bounded in the {{math|𝒢}}-topology.{{sfn | Schaefer |Wolff| 1999 | p=83}}</li>

</ul>

Line 242:

The weak-topology on {{math|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|L_𝜎(''X''; ''Y'')}} is metrizable; if in addition {{mvar|Y}} is separable then so is {{mvar|H}}.{{sfn | Schaefer |Wolff| 1999 | p=87}}

* So in particular, on every equicontinuous subset of {{math|L(''X''; ''Y'')}}, the topology of pointwise convergence is metrizable.</li>

<li>Let {{math|''Y''^''X''}} denote the space of all functions from {{mvar|X}} into {{mvar|Y}}. If {{math|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|''Y''^''X''}}.

* In addition, {{math|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|L(''X''; ''Y'')}} is bounded when {{math|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|L(''X''; ''Y'')}} are identical for all {{math|𝒢}}-topologies on {{math|L(''X''; ''Y'')}} such that {{math|𝒢}} is a family of bounded sets covering {{mvar|X}}.{{sfn | Schaefer |Wolff| 1999 | p=82}}</li>

</ul>

Line 253:

<li>The weak-closure of an equicontinuous subset of {{math|L(''X''; ''Y'')}} is equicontinuous.</li>

<li>If {{mvar|Y}} is locally convex, then the convex balanced hull of an equicontinuous subset of {{math|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|L(''X''; ''Y'')}} is equicontinuous.{{sfn | Schaefer |Wolff| 1999 | p=83}}</li>

<li>On an equicontinuous subset {{mvar|H}} of {{math|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 |Wolff| 1999 | p=83}}</li>

</ul>

Line 394:

{{Reflist}}

==Bibliography==

* {{Jarchow Locally Convex Spaces}}

* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} -->

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->

* {{cite book | last = Hogbe-Nlend | first = Henri | title = Bornologies and functional analysis | publisher = North-Holland Publishing Co. | ___location = Amsterdam | year = 1977 | pages = xii+144 | isbn = 0-7204-0712-5 | mr = 0500064}}

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer |Wolff| 1999 | p=}} -->

* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} -->

{{Functional Analysis}}