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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.
==
Throughout
<ol>
<li><math>T</math> is any non-empty set and <math>\mathcal{G}</math> is a non-empty collection of subsets of <math>T</math> [[Directed set|directed]] by subset inclusion (i.e. for any <math>G, H \in \mathcal{G}</math> there exists some <math>K \in \mathcal{G}</math> such that <math>G \cup H \subseteq K</math>).</li>
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<math display="block">\mathcal{U}(G, N) := \{ f \in F : f(G) \subseteq N \}.</math>
===
Henceforth assume that <math>G \in \mathcal{G}</math> and <math>N \in \mathcal{N}.</math>
<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>
<ul>
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</ul>
===
Then the family
<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 (
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>).
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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}}
{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=79-88}}{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
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}}
====
{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
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}}
===
If <math>Y</math> is [[locally convex]] then so is the <math>\mathcal{G}</math>-topology on <math>F</math> and if <math>\left(p_i\right)_{i \in I}</math> is a family of continuous seminorms generating this topology on <math>Y</math> then the <math>\mathcal{G}</math>-topology is induced by the following family of seminorms:
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as <math>G</math> varies over <math>\mathcal{G}</math> and <math>i</math> varies over <math>I</math>.{{sfn|Schaefer|Wolff|1999|p=81}}
If <math>Y</math> is [[Hausdorff space|Hausdorff]] and <math>T = \bigcup_{G \in \mathcal{G}} G</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff.{{sfn|Jarchow|1981|pp=43-55}}
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If <math>Y</math> is [[Hausdorff space|Hausdorff]] and <math>F</math> is the vector subspace of <math>Y^T</math> consisting of all continuous maps that are bounded on every <math>G \in \mathcal{G}</math> and if <math>\bigcup_{G \in \mathcal{G}} G</math> is dense in <math>T</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff.
A subset <math>H</math> of <math>F</math> is [[Bounded set (topological vector space)|bounded]] in the <math>\mathcal{G}</math>-topology if and only if for every <math>G \in \mathcal{G},</math> <math>H(G) = \bigcup_{h \in H} h(G)</math> is bounded in <math>Y.</math>{{sfn|Schaefer|Wolff|1999|p=81}}
===
If we let <math>\mathcal{G}</math> be the set of all finite subsets of <math>T</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is called the '''topology of pointwise convergence'''.
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If <math>X</math> is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and <math>C(X)</math> is the space of all real (or complex) valued continuous functions on <math>X,</math> the topology of pointwise convergence on <math>C(X)</math> is [[Metrizable TVS|metrizable]] if and only if <math>X</math> is countable.{{sfn|Jarchow|1981|pp=43-55}}
==
Throughout this section we will assume that <math>X</math> and <math>Y</math> are [[topological vector space]]s.
<math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by inclusion.
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.
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>
===
The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
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Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
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 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>
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}}
Some authors (e.g. Narici) require that <math>\mathcal{G}</math> satisfy the following condition, which implies, in particular, that <math>\mathcal{G}</math> is [[Directed set|directed]] by subset inclusion:
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If <math>\mathcal{G}</math> is a [[saturated family]] of [[Bounded set (topological vector space)|bounded]] subsets of <math>X</math> then these axioms are also satisfied.
===
'''Hausdorffness'''
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<li>If <math>X</math> is a Mackey space then <math>L_{\mathcal{G}}(X; Y)</math>is complete if and only if both <math>X^{\prime}_{\mathcal{G}}</math> and <math>Y</math> are complete.</li>
<li>If <math>X</math> is [[Barrelled space|barrelled]] then <math>L_{\mathcal{G}}(X; Y)</math> is Hausdorff and [[quasi-complete]].</li>
<li>Let <math>X</math> and <math>Y</math> be TVSs with <math>Y</math> [[quasi-complete]] and assume that (1) <math>X</math> is [[
<li>Let <math>X</math> be a [[bornological space]], <math>Y</math> a locally convex space, and <math>\mathcal{G}</math> a family of bounded subsets of <math>X</math> such that the range of every null sequence in <math>X</math> is contained in some <math>G \in \mathcal{G}.</math> If <math>Y</math> is [[quasi-complete]] (
</ul>
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</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, if <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then
<ul>
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</ul>
===
{| class="wikitable"
|-
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|}
====
By letting <math>\mathcal{G}</math> be the set of all finite subsets of <math>X,</math> <math>L(X; Y)</math> will have the '''weak topology on <math>L(X; Y)</math>''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and <math>L(X; Y)</math> with this topology is denoted by <math>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.
The weak-topology on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is [[Separable space|separable]] (
* So in particular, on every equicontinuous subset of <math>L(X; Y),</math> the topology of pointwise convergence is metrizable.</li>
<li>Let <math>Y^X</math> denote the space of all functions from <math>X</math> into <math>Y.</math> If <math>L(X; Y)</math> is given the topology of pointwise convergence then space of all linear maps (continuous or not) <math>X</math> into <math>Y</math> is closed in <math>Y^X</math>.
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</ul>
<ul>
<li>The weak-closure of an [[Equicontinuous linear maps|equicontinuous subset]] of <math>L(X; Y)</math> is equicontinuous.</li>
<li>If <math>Y</math> is locally convex, then the convex balanced hull of an equicontinuous subset of <math>L(X; Y)</math> is equicontinuous.</li>
<li>Let <math>X</math> and <math>Y</math> be TVSs and assume that (1) <math>X</math> is [[barreled space|barreled]], or else (2) <math>X</math> is a [[Baire space]] and <math>X</math> and <math>Y</math> are locally convex. Then every simply bounded subset of <math>L(X; Y)</math> is equicontinuous.{{sfn|Schaefer|Wolff|1999|p=83}}</li>
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</ul>
====
By letting <math>\mathcal{G}</math> be the set of all compact subsets of <math>X,</math> <math>L(X; Y)</math> will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and <math>L(X; Y)</math> with this topology is denoted by <math>L_c(X; Y)</math>.
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The topology of compact convergence on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is a [[Fréchet space]] or a [[LF-space]] and if <math>Y</math> is a [[Complete
<li>On [[Equicontinuous linear maps|equicontinuous subsets]] of <math>L(X; Y),</math> the following topologies coincide:
* The topology of pointwise convergence on a dense subset of <math>X,</math>
* The topology of pointwise convergence on <math>X,</math>
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</ul>
====
By letting <math>\mathcal{G}</math> be the set of all bounded subsets of <math>X,</math> <math>L(X; Y)</math> will have '''the topology of bounded convergence on <math>X</math>''' or '''the topology of uniform convergence on bounded sets''' and <math>L(X; Y)</math> with this topology is denoted by <math>L_b(X; Y)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
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The topology of bounded convergence on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is a [[bornological space]] and if <math>Y</math> is a [[Complete
<li>If <math>X</math> and <math>Y</math> are both normed spaces then the topology on <math>L(X; Y)</math> induced by the usual operator norm is identical to the topology on <math>L_b(X; Y)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* In particular, if <math>X</math> is a normed space then the usual norm topology on the continuous dual space <math>X^{\prime}</math> is identical to the topology of bounded convergence on <math>X^{\prime}</math>.</li>
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</ul>
==
{{Main|Polar topology}}
Throughout, we assume that <math>X</math> is a TVS.
===
If <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 <math>X</math> is a Hausdorff locally convex space), then a <math>\mathcal{G}</math>-topology on <math>X^{\prime}</math> (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a <math>\mathcal{G}</math>-topology.
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We list here some of the most common polar topologies.
===
Suppose that <math>X</math> is a TVS whose bounded subsets are the same as its weakly bounded subsets.
{| class="wikitable"
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|}
==
We will let <math>\mathcal{B}(X, Y; Z)</math> denote the space of separately continuous bilinear maps and <math>B(X, Y; Z)</math>denote the space of continuous bilinear maps, where <math>X, Y,</math> and <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>L(X; Y)</math> we can place a topology on <math>\mathcal{B}(X, Y; Z)</math> and <math>B(X, Y; Z)</math>.
Let <math>\mathcal{G}</math> (
Let <math>\mathcal{G} \times \mathcal{H}</math> denote the collection of all sets <math>G \times H</math> where <math>G \in \mathcal{G},</math> <math>H \in \mathcal{H}.</math>
We can place on <math>Z^{X \times Y}</math> the <math>\mathcal{G} \times \mathcal{H}</math>-topology, and consequently on any of its subsets, in particular on <math>B(X, Y; Z)</math>and on <math>\mathcal{B}(X, Y; Z)</math>.
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* <math>X</math> is normed and <math>Y</math> and <math>Z</math> the strong duals of reflexive Fréchet spaces.
===
{{Main|Injective tensor product}}
Suppose that <math>X, Y,</math> and <math>Z</math> are locally convex spaces and let <math>\mathcal{G}^{\prime}</math> and <math>\mathcal{H}^{\prime}</math> be the collections of [[Equicontinuous linear functionals|equicontinuous subsets]] of <math>X^{\prime}</math> and <math>X^{\prime}</math>, respectively.
Then the <math>\mathcal{G}^{\prime}-\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>
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* If <math>X</math> and <math>Y</math> are both normed (respectively, both Banach) then so is <math>\mathcal{B}_{\epsilon}\left(X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}\right)</math>
==
* {{annotated link|Bornological space}}
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* {{annotated link|Weak topology}}
==
{{reflist|group=note}}
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{{reflist}}
==
* {{Jarchow Locally Convex Spaces}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|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=}} -->
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{{Functional Analysis}}
{{DualityInLCTVSs}}
{{TopologicalVectorSpaces}}
[[Category:Topological vector spaces]]
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