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In [[mathematics]], particularly [[functional analysis]], spaces of [[linear map]]s between two [[vector space]]s can be endowed with a variety of [[Topology (structure)|topologies]]. Studying 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).
==Topologies of uniform convergence on arbitrary spaces of maps==
Throughout, the following is assumed:
<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>
<li><math>Y</math> is a [[topological vector space]] (not necessarily Hausdorff or locally convex).</li>
<li><math>\mathcal{N}</math> is a basis of neighborhoods of 0 in <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>
</ol>
==
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
<math display="block">\mathcal{U}(G, N) := \{f \in F : f(G) \subseteq N\}.</math>
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 (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>
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}}
{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=79-88}}{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
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 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.
Moreover,{{sfn|Narici|Beckenstein|2011|pp=19-45}}
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>if <math>M \subseteq N</math> then <math>\mathcal{U}(G, M) \subseteq \mathcal{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>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>
===Uniform structure===
{{See also|Uniform space}}
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
<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>
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}}
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}}
'''Nets and uniform convergence'''
Let <math>f \in F</math> and let <math>f_{\bull} = \left(f_i\right)_{i \in I}</math> be a [[Net (mathematics)|net]] in <math>F.</math> Then for any subset <math>G</math> of <math>T,</math> say that <math>f_{\bull}</math> '''converges uniformly to <math>f</math> on <math>G</math>''' if for every <math>N \in \mathcal{N}</math> there exists some <math>i_0 \in I</math> such that for every <math>i \in I</math> satisfying <math>i \geq i_0,I</math> <math>f_i - f \in \mathcal{U}(G, N)</math> (or equivalently, <math>f_i(g) - f(g) \in N</math> for every <math>g \in G</math>).{{sfn|Jarchow|1981|pp=43-55}}
{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
If <math>f \in F</math> and if <math>f_{\bull} = \left(f_i\right)_{i \in I}</math> is a net in <math>F,</math> then <math>f_{\bull} \to f</math> in the <math>\mathcal{G}</math>-topology on <math>F</math> if and only if for every <math>G \in \mathcal{G},</math> <math>f_{\bull}</math> converges uniformly to <math>f</math> on <math>G.</math>
}}
===Inherited properties===
'''Local convexity'''
<math display="block">p_{G,i}(f) := \sup_{x \in G} p_i(f(x)),</math>
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}}
'''Hausdorffness'''
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}}
'''Boundedness'''
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}}
===Examples of 𝒢-topologies===
'''Pointwise convergence'''
The topology of pointwise convergence on <math>F</math> is identical to the subspace topology that <math>F</math> inherits from <math>Y^T</math> when <math>Y^T</math> is endowed with the usual [[product topology]].
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}}
==𝒢-topologies on spaces of continuous linear maps==
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.
<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.
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===
'''Assumptions that guarantee a vector topology'''
The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
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.
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>
{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=371-423}}|math_statement=
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>:
<ol>
<li>all subsets of all finite unions of sets in <math>\mathcal{G}</math>;</li>
<li>all scalar multiples of all sets in <math>\mathcal{G}</math>;</li>
<li>all finite [[Minkowski sum]]s of sets in <math>\mathcal{G}</math>;</li>
<li>the [[Balanced set|balanced hull]] of every set in <math>\mathcal{G}</math>;</li>
<li>the closure of every set in <math>\mathcal{G}</math>;</li>
</ol>
and if <math>X</math> and <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{G}.</math></li>
</ol>
}}
'''Common assumptions'''
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:
:<math>\mathcal{G}</math> is assumed to be closed with respect to the formation of subsets of finite unions of sets in <math>\mathcal{G}</math> (i.e. every subset of every finite union of sets in <math>\mathcal{G}</math> belongs to <math>\mathcal{G}</math>).
Some authors (e.g. Trèves {{sfn|Trèves|2006|loc=Chapter 32}}) require that <math>\mathcal{G}</math> be directed under subset inclusion and that it satisfy the following condition:
:If <math>G \in \mathcal{G}</math> and <math>s</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that <math>s G \subseteq H.</math>
If <math>\mathcal{G}</math> is a [[bornology]] on <math>X,</math> which is often the case, then these axioms are satisfied.
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.
===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>
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 <math>F</math> is the vector subspace of <math>Y^T</math> consisting of all continuous linear maps that are bounded on every <math>G \in \mathcal{G},</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff if <math>Y</math> is Hausdorff and <math>\mathcal{G}</math> is total in <math>T.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}
'''Completeness'''
For the following theorems, suppose that <math>X</math> is a topological vector space and <math>Y</math> is a [[locally convex]] Hausdorff spaces and <math>\mathcal{G}</math> is a collection of bounded subsets of <math>X</math> that covers <math>X,</math> is directed by subset inclusion, and satisfies the following condition: if <math>G \in \mathcal{G}</math> and <math>s</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that <math>s G \subseteq H.</math>
<ul>
<li><math>L_{\mathcal{G}}(X; Y)</math> is [[Complete topological vector space|complete]] if
{{ordered list|
|<math>X</math> is locally convex and Hausdorff,
|<math>Y</math> is complete, and
|whenever <math>u : X \to Y</math> is a linear map then <math>u</math> restricted to every set <math>G \in \mathcal{G}</math> is continuous implies that <math>u</math> is continuous,
}}</li>
<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 [[Barreled space|barreled]], or else (2) <math>X</math> is a [[Baire space]] and <math>X</math> and <math>Y</math> are locally convex. If <math>\mathcal{G}</math> covers <math>X</math> then every closed [[Equicontinuous linear maps|equicontinuous subset]] of <math>L(X; Y)</math> is complete in <math>L_{\mathcal{G}}(X; Y)</math> and <math>L_{\mathcal{G}}(X; Y)</math> is quasi-complete.{{sfn|Schaefer|Wolff|1999|p=83}}</li>
<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]] (respectively, [[Complete topological vector space|complete]]) then so is <math>L_{\mathcal{G}}(X; Y)</math>.{{sfn|Schaefer|Wolff|1999|p=117}}</li>
</ul>
'''Boundedness'''
Let <math>X</math> and <math>Y</math> be topological vector spaces and <math>H</math> be a subset of <math>L(X; Y).</math>
Then the following are equivalent:{{sfn|Schaefer|Wolff|1999|p=81}}
<ol>
<li><math>H</math> is [[Bounded set (topological vector space)|bounded]] in <math>L_{\mathcal{G}}(X; Y)</math>;</li>
<li>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}}</li>
<li>For every neighborhood <math>V</math> of the origin in <math>Y</math> the set <math>\bigcap_{h \in H} h^{-1}(V)</math> [[Absorbing set|absorbs]] every <math>G \in \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, if <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then
<ul>
<li>if <math>H</math> is bounded in <math>L_{\sigma}(X; Y)</math> (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 <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>
<li> if <math>X</math> is [[Quasi-complete space|quasi-complete]] (meaning that closed and bounded subsets are complete), then the bounded subsets of <math>L(X; Y)</math> are identical for all <math>\mathcal{G}</math>-topologies where <math>\mathcal{G}</math> is any family of bounded subsets of <math>X</math> covering <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>
<li></li>
</ul>
===Examples===
{| class="wikitable"
|-
! <math>\mathcal{G} \subseteq \wp(X)</math> ("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
! Alternative name
|-
| finite subsets of <math>X</math>
| <math>L_{\sigma}(X; Y)</math>
| pointwise/simple convergence
| topology of simple convergence
|-
| precompact subsets of <math>X</math>
|
| precompact convergence
|
|-
| compact convex subsets of <math>X</math>
| <math>L_{\gamma}(X; Y)</math>
| compact convex convergence
|
|-
| compact subsets of <math>X</math>
| <math>L_c(X; Y)</math>
| compact convergence
|
|-
| bounded subsets of <math>X</math>
| <math>L_b(X; Y)</math>
| bounded convergence
| strong topology
|}
====The topology of pointwise convergence====
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.
A subset of <math>L(X; Y)</math> is called '''simply bounded''' or '''weakly bounded''' if it is bounded in <math>L_{\sigma}(X; Y)</math>.
The weak-topology on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is [[Separable space|separable]] (that is, it has a countable dense subset) and if <math>Y</math> is a metrizable topological vector space then every [[Equicontinuous linear maps|equicontinuous subset]] <math>H</math> of <math>L_{\sigma}(X; Y)</math> is metrizable; if in addition <math>Y</math> is separable then so is <math>H.</math>{{sfn|Schaefer|Wolff|1999|p=87}}
* 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>.
* In addition, <math>L(X; Y)</math> is dense in the space of all linear maps (continuous or not) <math>X</math> into <math>Y.</math></li>
<li>Suppose <math>X</math> and <math>Y</math> are locally convex. Any simply bounded subset of <math>L(X; Y)</math> is bounded when <math>L(X; Y)</math> has the topology of uniform convergence on convex, [[balanced set|balanced]], bounded, complete subsets of <math>X.</math> If in addition <math>X</math> is [[quasi-complete]] then the families of bounded subsets of <math>L(X; Y)</math> are identical for all <math>\mathcal{G}</math>-topologies on <math>L(X; Y)</math> such that <math>\mathcal{G}</math> is a family of bounded sets covering <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>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>
<li>On an equicontinuous subset <math>H</math> of <math>L(X; Y),</math> the following topologies are identical: (1) topology of pointwise convergence on a total subset of <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====
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>.
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 topological vector space|complete]] locally convex Hausdorff space then <math>L_c(X; Y)</math> is complete.</li>
<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>
* The topology of compact convergence.
* The topology of precompact convergence.</li>
<li>If <math>X</math> is a [[Montel space]] and <math>Y</math> is a topological vector space, then <math>L_c(X; Y)</math> and <math>L_b(X; Y)</math> have identical topologies.</li>
</ul>
====Topology of bounded convergence====
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}}
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 topological vector space|complete]] locally convex Hausdorff space then <math>L_b(X; Y)</math> is complete.</li>
<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>
<li>Every equicontinuous subset of <math>L(X; Y)</math> is bounded in <math>L_b(X; Y)</math>.</li>
</ul>
==Polar topologies==
{{Main|Polar topology}}
Throughout, we assume that <math>X</math> is a TVS.
===𝒢-topologies versus polar topologies===
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.
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if <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 <math>X</math>" is stronger than the notion of "<math>\sigma\left(X, X^{\prime}\right)</math>-bounded in <math>X</math>" (i.e. bounded in <math>X</math> implies <math>\sigma\left(X, X^{\prime}\right)</math>-bounded in <math>X</math>) so that a <math>\mathcal{G}</math>-topology on <math>X^{\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>-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 <math>X</math> is a TVS whose bounded subsets are the same as its weakly bounded subsets.
'''Notation''': If <math>\Delta(Y, X)</math> denotes a polar topology on <math>Y</math> then <math>Y</math> endowed with this topology will be denoted by <math>Y_{\Delta(Y, X)}</math> or simply <math>Y_{\Delta}</math> (e.g. for <math>\sigma(Y, X)</math> we would have <math>\Delta = \sigma</math> so that <math>Y_{\sigma(Y, X)}</math> and <math>Y_{\sigma}</math> all denote <math>Y</math> with endowed with <math>\sigma(Y, X)</math>).
{| class="wikitable"
|-
! ><math>\mathcal{G} \subseteq \wp(X)</math><br/>("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
! Alternative name
|-
| finite subsets of <math>X</math>
| <math>\sigma(Y, X)</math><br/><math>s(Y, X)</math>
| pointwise/simple convergence
| [[Weak topology|weak/weak* topology]]
|-
| <math>\sigma(X, Y)</math>-compact [[Absolutely convex set|disk]]s
| <math>\tau(Y, X)</math>
|
| [[Mackey topology]]
|-
| <math>\sigma(X, Y)</math>-compact convex subsets
| <math>\gamma(Y, X)</math>
| compact convex convergence
|
|-
| <math>\sigma(X, Y)</math>-compact subsets<br/>(or balanced <math>\sigma(X, Y)</math>-compact subsets)
| <math>c(Y, X)</math>
| compact convergence
|
|-
| <math>\sigma(X, Y)</math>-bounded subsets
| <math>b(Y, X)</math><br/><math>\beta(Y, X)</math>
| bounded convergence
| [[Strong dual space|strong topology]]
|}
==𝒢-ℋ topologies on spaces of bilinear maps==
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> (respectively, <math>\mathcal{H}</math>) be a family of subsets of <math>X</math> (respectively, <math>Y</math>) containing at least one non-empty set.
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>.
This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math>-topology''' or as the '''topology of uniform convergence on the products <math>G \times H</math> of <math>\mathcal{G} \times \mathcal{H}</math>'''.
However, as before, this topology is not necessarily compatible with the vector space structure of <math>\mathcal{B}(X, Y; Z)</math> or of <math>B(X, Y; Z)</math> without the additional requirement that for all bilinear maps, <math>b</math> in this space (that is, in <math>\mathcal{B}(X, Y; Z)</math> or in <math>B(X, Y; Z)</math>) and for all <math>G \in \mathcal{G}</math> and <math>H \in \mathcal{H},</math> the set <math>b(G, H)</math> is bounded in <math>X.</math>
If both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consist of bounded sets then this requirement is automatically satisfied if we are topologizing <math>B(X, Y; Z)</math> but this may not be the case if we are trying to topologize <math>\mathcal{B}(X, Y; Z)</math>.
The <math>\mathcal{G}-\mathcal{H}</math>-topology on <math>\mathcal{B}(X, Y; Z)</math> will be compatible with the vector space structure of <math>\mathcal{B}(X, Y; Z)</math> if both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consists of bounded sets and any of the following conditions hold:
* <math>X</math> and <math>Y</math> are barrelled spaces and <math>Z</math> is locally convex.
* <math>X</math> is a [[F-space]], <math>Y</math> is metrizable, and <math>Z</math> is Hausdorff, in which case <math>\mathcal{B}(X, Y; Z) = B(X, Y; Z).</math>
* <math>X, Y,</math> and <math>Z</math> are the strong duals of reflexive Fréchet spaces.
* <math>X</math> is normed and <math>Y</math> and <math>Z</math> the strong duals of reflexive Fréchet spaces.
===The ε-topology===
{{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>
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 <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 <math>X</math> and <math>Y.</math>
In fact, if <math>X</math> and <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 <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then <math>\mathcal{B}_{\varepsilon}\left(X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}\right)</math> is complete if and only if both <math>X</math> and <math>Y</math> are complete.
* 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>
==See also==
* {{annotated link|Bornological space}}
* {{annotated link|Bounded linear operator}}
* {{annotated link|Dual system}}
* {{annotated link|Dual topology}}
* {{annotated link|List of topologies}}
* {{annotated link|Modes of convergence}}
* {{annotated link|Operator norm}}
* {{annotated link|Polar topology}}
* {{annotated link|Strong dual 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}}
==Bibliography==
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn|Hogbe-Nlend|1977|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{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=}} -->
{{Functional analysis}}
{{Duality and spaces of linear maps}}
{{Topological vector spaces}}
[[Category:Functional analysis]]
[[Category:Topological vector spaces]]
[[Category:Topology of function spaces]]
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