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In [[mathematics]], aparticularly [[linearfunctional mapanalysis]], isspaces aof [[functionlinear (mathematics)|mappingmap]] {{math|''V'' → ''W''}}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, the following is assumed:
Suppose that ''T'' is any set and that <math>\mathcal{G}</math> is a collection of subsets of ''T'' directed by inclusion. Suppose in addition that ''Y'' is a [[topological vector space]] (not necessarily Hausdorff or locally convex) and that <math>\mathcal{N}</math> is a basis of neighborhoods of 0 in ''Y''. Then the set of all functions from ''T'' into ''Y'', <math>Y^T</math>, can be given a unique translation-invariant topology by defining a basis of neighborhoods of 0 in <math>Y^T</math>, to be
<ol>
:<math>\mathcal{U}(G, N) = \{f \in Y^T : f(G) \subseteq N\}</math>
<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>
as ''G'' and ''N'' range over all <math>G \in \mathcal{G}</math> and <math>N \in \mathcal{N}</math>. This topology does not depend on the 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'''.<ref name="Schaefer (1970) p. 79">Schaefer (1970) p. 79</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 ''T'' (and ''T'' is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of ''T''. A set <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>Y^T</math>.<ref name="Schaefer (1970) p. 79" />
<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>
===𝒢-topology===
However, the <math>\mathcal{G}</math>-topology on <math>Y^T</math> is not necessarily compatible with the vector space structure of <math>Y^T</math> or of any of its vector subspaces (that is, it is not necessarily a topological vector space topology on <math>Y^T</math>). Suppose that ''F'' is a vector subspace <math>Y^T</math> so that it inherits the subspace topology from <math>Y^T</math>. Then the <math>\mathcal{G}</math>-topology on ''F'' is compatible with the vector space structure of ''F'' if and only if for every <math>G \in \mathcal{G}</math> and every ''f'' ∈ ''F'', ''f''(''G'') is bounded in ''Y''.<ref name="Schaefer (1970) p. 79" />
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
If ''Y'' is locally convex then so is the <math>\mathcal{G}</math>-topology on <math>Y^T</math> and if <math>(p_{\alpha})</math> is a family of continuous seminorms generating this topology on ''Y'' then the <math>\mathcal{G}</math>-topology is induced by the following family of seminorms: <math>p_{G, \alpha}(f) = \sup_{x \in G} p_{\alpha}(f(x))</math>, as ''G'' varies over <math>\mathcal{G}</math> and <math>\alpha</math> varies over all indices.<ref name="Schaefer (1970) p. 81">Schaefer (1970) p. 81</ref> If ''Y'' is Hausdorff and ''T'' is a topological space such that <math>\bigcup_{G \in \mathcal{G}} G</math> is dense in ''T'' then the <math>\mathcal{G}</math>-topology on subspace of <math>Y^T</math> consisting of all continuous maps is Hausdorff. If the topological space ''T'' is also a topological vector space, then the condition that <math>\bigcup_{G \in \mathcal{G}} G</math> be dense in ''T'' can be replaced by the weaker condition that the linear span of this set be dense in ''T'', in which case we say that this set is '''total in ''T'''''.<ref name="Schaefer (1970) p. 80">Schaefer (1970) p. 80</ref>
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
Let ''H'' be a subset of <math>Y^T</math>. Then ''H'' is bounded in the <math>\mathcal{G}</math>-topology if and only if for every <math>G \in \mathcal{G}</math>, <math>\cup_{u \in H} u(G)</math> is bounded in ''Y''.<ref name="Schaefer (1970) p. 81">Schaefer (1970) p. 81</ref>
<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>
==Spaces of continuous linear maps==
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}}
Throughout this section we will assume that ''X'' and ''Y'' are [[topological vector space]]s and we will let ''L''(''X'', ''Y''), denote the vector space of all continuous linear maps from ''X'' and ''Y''. If ''L''(''X'', ''Y'') 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 <math>\mathcal{G}</math>-topology on ''L''(''X'', ''Y'') is compatible with the vector space structure of ''L''(''X'', ''Y'') if and only if for all <math>G \in \mathcal{G}</math> and all ''f'' ∈ ''L''(''X'', ''Y'') the set ''f''(''G'') is bounded in ''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>\mathcal{G}</math> consists of (von-Neumann) bounded subsets of ''X''.
{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=79-88}}{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
Often, <math>\mathcal{G}</math> is required to satisfy the following two axioms:
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>
|-
}}
| align="right" | <math>\mathcal{G}_1</math>: || If <math>G_1, G_2 \in \mathcal{G}</math> then there exists a <math>G \in \mathcal{G}</math> such that <math>G_1 \cup G_2 \subseteq G</math>.
|-
| align="right" | <math>\mathcal{G}_2</math>: || If <math>G_1 \in \mathcal{G}</math> and <math>\lambda</math> is a scalar then there exists a <math>G \in \mathcal{G}</math> such that <math>\lambda G_1 \subseteq G</math>.
|}
'''Properties'''
If <math>\mathcal{G}</math> is a [[bornology]] on ''X''. which is often the case, then these two axioms are satisfied.
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>
===Properties===
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
====Completeness====
<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}}
For the following theorems, suppose that ''X'' is a topological vector space and ''Y'' is a locally convex Hausdorff spaces and <math>\mathcal{G}</math> is a collection of bounded subsets of ''X'' that satisfies axioms <math>\mathcal{G}_1</math> and <math>\mathcal{G}_2</math> and forms a covering of ''X''.
* <math display=block>L_\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G}, M) \cap \mathcal{U}(XH, YN)</math> is complete if
which implies:
{{ordered list|
<ul>
| ''X'' is locally convex and Hausdorff,
<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>
| ''Y'' is complete, and
| whenever<li>if <math>u : XG \tosubseteq YH</math> is a linear map then ''u'' restristed to every set <math>G\mathcal{U}(H, N) \insubseteq \mathcal{U}(G}, N).</math> is continuous implies that ''u'' is continuous,</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>
* If ''X'' is a Mackey space then <math>L_{\mathcal{G}}(X, Y)</math> is complete if and only if both <math>X^*_{\mathcal{G}}</math> and ''Y'' are complete.
* If ''X'' is [[Barrelled space|barrelled]] then <math>L_{\mathcal{G}}(X, Y)</math> is Hausdorff and quasi-complete, which means that every closed and bounded set is complete.
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>
====Boundedness====
===Uniform structure===
Let ''X'' and ''Y'' be topological vector space and ''H'' be a subset of ''L''(''X'', ''Y''). Then the following are equivalent:<ref name="Schaefer (1970) p. 81">Schaefer (1970) p. 81</ref>
{{See also|Uniform space}}
* ''H'' is bounded in <math>L_{\mathcal{G}}(X, Y)</math>,
* For every <math>G \in \mathcal{G}</math>, <math>\cup_{u \in H} u(G)</math> is bounded in ''Y'',
* For every neighborhood of 0, ''V'', in ''Y'' the set <math>\cap_{u \in H} u^{-1}(V)</math> absorbs every <math>G \in \mathcal{G}</math>.
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
Furthermore,
<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>
* If ''X'' and ''Y'' are locally convex Hausdorff space and if ''H'' is bounded in <math>L_{\sigma}(X, Y)</math> (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 ''X''.<ref name="Schaefer (1970) p. 82">Schaefer (1970) p. 82</ref>
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}}
* If ''X'' and ''Y'' are locally convex Hausdorff spaces and if ''X'' is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of ''L(X, Y)'' are identical for all <math>\mathcal{G}</math>-topologies where <math>\mathcal{G}</math> is any family of bounded subsets of ''X'' covering ''X''.<ref name="Schaefer (1970) p. 82" />
* IfThe {{em|<math>\mathcal{G}</math>-convergence isuniform any collection of bounded subsets of ''X'' whose unionstructure}} is total in ''X'' thenthe everyleast equicontinuousupper subsetbound of ''L(X,all Y)''<math>G</math>-convergence isuniform boundedstructures in theas <math>G \in \mathcal{G}</math>-topology.<ref name="Schaeferranges (1970)over p. 83"<math>Schaefer (1970) p\mathcal{G}. 83</refmath>{{sfn|Grothendieck|1973|pp=1-13}}
'''Nets and uniform convergence'''
===Examples===
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}}
====The topology of pointwise convergence ''L''<sub>''σ''</sub>(''X'', ''Y'')====
{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
By letting <math>\mathcal{G}</math> be the set of all finite subsets of ''X'', ''L(X, Y)'' will have the '''weak topology on ''L(X, Y)''''' or '''the topology of pointwise convergence''' and ''L(X, Y)'' with this topology is denoted by <math>L_{\sigma}(X, Y)</math>
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===
The weak-topology on ''L(X, Y)'' has the following properties:
* The weak-closure of an equicontinuous subset of ''L(X, Y)'' is equicontinuous.
* If ''Y'' is locally convex, then the convex balanced hull of an equicontinuous subset of <math>L(X, Y)</math> is equicontinuous.
* If ''A ⊆ X'' is a countable dense subset of a topological vector space ''X'' and if ''Y'' is a metrizable topological vector space then <math>L_{\sigma}(X, Y)</math> is metrizable.
** So in particular, on every equicontinuous subset of ''L(X, Y)'', the topology of pointwise convergence is metrizable.
* Let <math>Y^X</math> denote the space of all functions from ''X'' into ''Y''. If <math>F(X, Y)</math> is given the topology of pointwise convergence then space of all linear maps (continuous or not) ''X'' into ''Y'' is closed in <math>Y^X</math>.
** In addition, ''L(X, Y)'' is dense in the space of all linear maps (continuous or not) ''X'' into ''Y''.
'''Local convexity'''
====Compact-convex convergence ''L''<sub>''γ''</sub>(''X'', ''Y'')====
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:
By letting <math>\mathcal{G}</math> be the set of all compact convex subsets of ''X'', ''L(X, Y)'' will have '''the topology of compact convex convergence''' or '''the topology of uniform convergence on compact convex sets''' ''L(X, Y)'' with this topology is denoted by <math>L_{\gamma}(X, Y)</math>.
<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'''
====Compact convergence ''L''<sub>''c''</sub>(''X'', ''Y'')====
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}}
By letting <math>\mathcal{G}</math> be the set of all compact subsets of ''X'', ''L(X, Y)'' will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and ''L(X, Y)'' with this topology is denoted by <math>L_{c}(X, Y)</math>.
Suppose that <math>T</math> is a topological space.
The topology of bounded convergence on ''L(X, Y)'' has the following properties:
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.
* If ''X'' is a [[Fréchet space]] or a [[LF-space]] and if ''Y'' is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then <math>L_{c}(X, Y)</math> is complete.
* On equicontinuous subsets of ''L(X, Y)'', the following topologies coincide:
** The topology of pointwise convergence on a dense subset of ''X'',
** The topology of pointwise convergence on ''X'',
** The topology of compact convergence.
* If ''X'' is a [[Montel space]] and ''Y'' is a topological vector space, then <math>L_{c}(X, Y)</math> and <math>L_{b}(X, Y)</math> have identical topologies.
'''Boundedness'''
====Strong dual topology ''L''<sub>''b''</sub>(''X'', ''Y'')====
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}}
By letting <math>\mathcal{G}</math> be the set of all bounded subsets of ''X'', ''L(X, Y)'' will have '''the topology of bounded convergence on ''X''''' or '''the topology of uniform convergence on bounded sets''' and ''L(X, Y)'' with this topology is denoted by <math>L_{b}(X, Y)</math>.
===Examples of 𝒢-topologies===
The topology of bounded convergence on ''L(X, Y)'' has the following properties:
* If ''X'' is a [[bornological space]] and if ''Y'' is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then <math>L_{b}(X, Y)</math> is complete.
* If ''X'' and ''Y'' are both normed spaces then <math>L_{b}(X, Y)</math> is a normed space with the usual operator norm.
* Every equicontinuous subset of ''L(X, Y)'' is bounded in <math>L_{b}(X, Y)</math>.
'''Pointwise convergence'''
==G-topologies on the continuous dual induced by ''X''==
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'''.
The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space ''X'' over the field <math>\mathcal{F}</math> (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space <math>L(X, \mathcal{F})</math> and is denoted by <math>X^*</math> and sometimes by <math>X'</math>. Given <math>\mathcal{G}</math>, a set of subsets of ''X'', we can apply all of the preceding to this space by using <math>Y = \mathcal{F}</math> and in this case <math>X^*</math> with this <math>\mathcal{G}</math>-topology is denoted by <math>X^*_{\mathcal{G}}</math>, so that in particular we have the following basic properties:
*The A basistopology of neighborhoods ofpointwise 0convergence foron <math>X^*_{\mathcal{G}}F</math> is formed,identical asto the subspace topology that <math>GF</math> variesinherits overfrom <math>\mathcal{G}Y^T</math>, by the [[polar set]]swhen <math>GY^\circT</math> :=is \{x'endowed \inwith X^*the :usual \sup_{x[[product \in G} |\langle x', x \rangle | \le 1\}</math>topology]].
** A [[Filter (mathematics)|filter]] <math>F'</math> on <math>X^*</math> converges to an element <math>x' \in X^*</math> in the <math>\mathcal{G}</math>-topology on <math>X^*</math> if <math>F'</math> uniformly to <math>x'</math> on each <math>G \in \mathcal{G}</math>.
** If ''G ⊆ X'' is bounded then <math>G^\circ</math> is absorbing, so <math>\mathcal{G}</math> usually consists of bounded subsets of ''X''.
* <math>X^*_{\mathcal{G}}</math> is locally convex,
* If <math>\bigcup_{G \in \mathcal{G}} G</math> is dense in ''X'' then <math>X^*_{\mathcal{G}}</math> is Hausdorff.
* If <math>\bigcup_{G \in \mathcal{G}} G</math> covers ''X'' then the canonical map from ''X'' into <math>(X^*_{\mathcal{G}})^*</math> is well-defined. That is, for all <math>x \in X</math> the evaluation functional on <math>X^*</math> (i.e. <math>x' \in X^* \mapsto \langle x', x \rangle</math>) is continuous on <math>X^*_{\mathcal{G}}</math>.
** If in addition <math>X^*</math> separates points on ''X'' then the canonical map of ''X'' into <math>(X^*_{\mathcal{G}})^*</math> is an injection.
* Suppose that ''X'' and ''Y'' are two topological vector spaces and <math>u : E \to F</math> is a continuous linear map. Suppose that <math>\mathcal{G}</math> and <math>\mathcal{H}</math> are collections of bounded subsets of ''X'' and ''Y'', respectively, that both satisfy axioms <math>\mathcal{G}_1</math> and <math>\mathcal{G}_2</math>. Then <math>u</math>'s [[transpose]], <math>{}^tu : Y^*_{\mathcal{H}} \to X^*_{\mathcal{G}}</math> is continuous if for every <math>G \in \mathcal{G}</math> there is a <math>H \in \mathcal{H}</math> such that ''u(G) ⊆ H''.<ref>Treves pp. 199–200</ref>
** In particular, the transpose of <math>u</math> is continuous if <math>X^*</math> carries the <math>\sigma(X^*, X)</math> (respectively, <math>\gamma(X^*, X)</math>, <math>c(X^*, X)</math>, <math>b(X^*, X)</math>) topology and <math>Y^*</math> carry any topology stronger than the <math>\sigma(Y^*, Y)</math> topology (respectively, <math>\gamma(Y^*, Y)</math>, <math>c(Y^*, Y)</math>, <math>b(Y^*, Y)</math>).
* If ''X'' is a locally convex Hausdorff topological vector space over the field <math>\mathcal{F}</math> and <math>\mathcal{G}</math> is a collection of bounded subsets of ''X'' that satisfies axioms <math>\mathcal{G}_1</math> and <math>\mathcal{G}_2</math> then the bilinear map <math>X \times X^*_{\mathcal{G}} \to \mathcal{F}</math> defined by <math>(x, x') \mapsto \langle x', x \rangle = x'(x)</math> is continuous if and only if ''X'' is normable and the <math>\mathcal{G}</math>-topology on <math>X^*</math> is the strong dual topology <math>b(X^*, X)</math>.
* Suppose that ''X'' is a Fréchet space and <math>\mathcal{G}</math> is a collection of bounded subsets of ''X'' that satisfies axioms <math>\mathcal{G}_1</math> and <math>\mathcal{G}_2</math>. If <math>\mathcal{G}</math> contains all compact subsets of ''X'' then <math>X^*_{\mathcal{G}}</math> is complete.
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}}
===Examples===
==𝒢-topologies on spaces of continuous linear maps==
====The weak topology ''σ''(''X''<sup>*</sup>, ''X'') or the weak* topology====
Throughout this section we will assume that <math>X</math> and <math>Y</math> are [[topological vector space]]s.
By letting <math>\mathcal{G}</math> be the set of all finite subsets of ''X'', <math>X^*</math> will have the '''weak topology on <math>X^*</math>''' more commonly known as the '''weak* topology''' or '''the topology of pointwise convergence''', which is denoted by <math>\sigma(X^*, X)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{\sigma}</math> or by <math>X^*_{\sigma(X^*, X)}</math> if there may be ambiguity.
<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.
The <math>\sigma(X^*, X)</math> topology has the following properties:
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>
* '''Theorem''' (S. Banach): Suppose that ''X'' and ''Y'' are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that <math>u : X \to Y</math> is a continuous linear map. Then <math>u</math> is surjective if and only if the transpose of <math>u</math>, <math>{}^t u : Y^* \to X^*</math>, is one-to-one and the range of <math>{}^t u</math> is weakly closed in <math>X^*_{\sigma(X^*, X)}</math>.
* Suppose that ''X'' and ''Y'' are Fréchet spaces, <math>Z</math> is a Hausdorff locally convex space and that <math>u : X^*_{\sigma} \times Y^*_{\sigma} \to Z^*_{\sigma}</math> is a separately-continuous bilinear map. Then <math>u : X^*_{b} \times Y^*_{b} \to Z^*_{b}</math> is continuous.
** In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
* <math>X^*_{\sigma(X^*, X)}</math> is normable if and only if ''X'' is finite-dimensional.
* When ''X'' is infinite-dimensional the <math>\sigma(X^*, X)</math> topology on <math>X^*</math> is strictly less fine than the strong dual topology <math>b(X^*, X)</math>.
* The <math>\sigma(X^*, X)</math>-closure of the convex balanced hull of an equicontinuous subset of <math>X^*</math> is equicontinuous and <math>\sigma(X^*, X)</math>-compact.
* Suppose that ''X'' is a locally convex Hausdorff space and that <math>\hat{X}</math> is its completion. If <math>X \neq \hat{X}</math> then <math>\sigma(X^*, \hat{X})</math> is strictly finer than <math>\sigma(X^*, X)</math>.
* Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the <math>\sigma(X^*, X)</math> topology.
===Assumptions on 𝒢===
====Compact-convex convergence ''γ''(''X''<sup>*</sup>, ''X'')====
'''Assumptions that guarantee a vector topology'''
By letting <math>\mathcal{G}</math> be the set of all compact convex subsets of ''X'', <math>X^*</math> will have '''the topology of compact convex convergence''' or '''the topology of uniform convergence on compact convex sets''', which is denoted by <math>\gamma(X^*, X)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{\gamma}</math> or by <math>X^*_{\gamma(X^*, X)}</math>.
* If ''X'' is a Fréchet space then the topologies <math>\gamma(X^*, X) = c(X^*, X)</math>.
* (<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>.
====Compact convergence ''c''(''X''<sup>*</sup>, ''X'')====
The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
By letting <math>\mathcal{G}</math> be the set of all compact subsets of ''X'', <math>X^*</math> will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''', which is denoted by <math>c(X^*, X)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{c}</math> or by <math>X^*_{c(X^*, X)}</math>.
The next assumption will guarantee that the sets <math>\mathcal{U}(G, N)</math> are [[Balanced set|balanced]].
* If ''X'' is a [[Fréchet space]] or a [[LF-space]] then <math>c(X^*, X)</math> is complete.
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
* Suppose that ''X'' is a metrizable topological vector space and that <math>W' \subseteq X^*</math>. If the intersection of <math>W'</math> with every equicontinuous subset of <math>X^*</math> is weakly-open, then <math>W'</math> is open in <math>c(X^*, X)</math>.
* (<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.
====Precompact convergence====
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>
By letting <math>\mathcal{G}</math> be the set of all precompact subsets of ''X'', <math>X^*</math> will have '''the topology of precompact convergence''' or '''the topology of uniform convergence on precompact sets'''.
* '''Alaoglu–Bourbaki Theorem''': An equicontinuous subset ''K'' of <math>X^*</math> has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on ''K'' coincides with the <math>\sigma(X^*, X)</math> topology.
* (<math>G \in \mathcal{G}</math> are bounded): <math>\mathcal{G}</math> is assumed to consist entirely of bounded subsets of <math>X.</math>
====Mackey topology ''τ''(''X''<sup>*</sup>, ''X'')====
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>
By letting <math>\mathcal{G}</math> be the set of all convex balanced weakly compact subsets of ''X'', <math>X^*</math> will have the '''Mackey topology on <math>X^*</math>''' or '''the topology of uniform convergence on convex balanced weakly compact sets''', which is denoted by <math>\tau(X^*, X)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{\tau(X^*, X)}</math>.
{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=371-423}}|math_statement=
====Strong dual topology ''b''(''X''<sup>*</sup>, ''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>:
<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:
By letting <math>\mathcal{G}</math> be the set of all bounded subsets of ''X'', <math>X^*</math> will have '''the topology of bounded convergence on ''X''''' or '''the topology of uniform convergence on bounded sets''' or the '''strong dual topology on <math>X^*</math>''', which is denoted by <math>b(X^*, X)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{b}</math> or by <math>X^*_{b(X^*, X)}</math>. Due to its importance, the continuous dual space of <math>X^*_{b}</math>, which is commonly denoted by <math>X^{**}</math> so that <math>(X^*_{b})^* = X^{**}</math>.
<ol start=6>
<li>the closed [[Absolutely convex|convex balanced hull]] of every set in <math>\mathcal{G}.</math></li>
</ol>
}}
'''Common assumptions'''
The <math>b(X^*, X)</math> topology has the following properties:
* If ''X'' is locally convex, then this topology is finer than all other <math>\mathcal{G}</math>-topologies on <math>X^*</math> when considering only <math>\mathcal{G}</math>'s whose sets are subsets of ''X''.
* If ''X'' is a [[bornological space]] (ex: [[metrizable]] or [[LF-space]]) then <math>X^*_{b(X^*, X)}</math>is complete.
* If ''X'' is a normed space then the strong dual topology on <math>X^*</math> may be defined by the norm <math>\|x'\| = \sup_{x \in X,, \|x\| = 1} | \langle x', x \rangle |</math>, where <math>x' \in X^*</math>.<ref>Treves, p. 198</ref>
* If ''X'' is a [[LF-space]] that is the inductive limit of the sequence of space <math>X_k</math> (for <math>k = 0,1 \dots</math>) then <math>X^*_{b(X^*, X)}</math> is a [[Fréchet space]] if and only if all <math>X_k</math> are normable.
* If ''X'' is a [[Montel space]] then
** <math>X^*_{b(X^*, X)}</math> has the Heine–Borel property (i.e. every closed and bounded subset of <math>X^*_{b(X^*, X)}</math> is compact in <math>X^*_{b(X^*, X)}</math>)
** On bounded subsets of <math>X^*_{b(X^*, X)}</math>, the strong and weak topologies coincide (and hence so do all other topologies finer than <math>\sigma(X^*, X)</math> and coarser than <math>b(X^*, X)</math>).
** Every weakly convergent sequence in <math>X^*</math> is strongly convergent.
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:
====Mackey topology ''τ''(''X''<sup>*</sup>, ''X''<sup>**</sup>)====
:<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:
By letting <math>\mathcal{G''}</math> be the set of all convex balanced weakly compact subsets of <math>X^{**} = (X^*_{b})^*</math>, <math>X^*</math> will have the ''Mackey topology on <math>X^*</math> induced by <math>X^{**}</math>''' or '''the topology of uniform convergence on convex balanced weakly compact subsets of <math>X^{**}</math>''', which is denoted by <math>\tau(X^*, X^{**})</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{\tau(X^*, X^{**})}</math>.
*:If This<math>G topology\in is\mathcal{G}</math> finer thanand <math>b(X^*, X)s</math> andis hencea finerscalar thanthen there exists a <math>H \tau(X^*,in X)\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.
====Other examples=Properties===
'''Hausdorffness'''
Other <math>\mathcal{G}</math>-topologies on <math>X^*</math> include
* The topology of uniform convergence on convex balanced complete bounded subsets of ''X''.
* The topology of uniform convergence on convex balanced infracomplete bounded subsets of ''X''.
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>
==G-topologies on ''X'' induced by the continuous dual==
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}}
There is a canonical map from ''X'' into <math>(X^*_{\sigma})^*</math> which maps an element <math>x \in X</math> to the following map: <math>x' \in X^* \mapsto \langle x', x \rangle</math>. By using this canonical map we can identify ''X'' as being contained in the continuous dual of <math>X^*_{\sigma}</math> i.e. contained in <math>(X^*_{\sigma})^*</math>. In fact, this canonical map is ''onto'', which means that <math>X = (X^*_{\sigma})^*</math> so that we can through this canonical isomorphism think of ''X'' as the continuous dual space of <math>X^*_{\sigma}</math>. Note that it is a common convention that if an equal sign appears between two sets which are clearly not equal, then the equality really means that the sets are isomorphic through some canonical map.
'''Completeness'''
Since we are now regarding ''X'' as the continuous dual space of <math>X^*_{\sigma}</math>, we can look at sets of subsets of <math>X^*_{\sigma}</math>, say <math>\mathcal{G'}</math> and construct a dual space topology on the dual of <math>X^*_{\sigma}</math>, which is ''X''. * A basis of neighborhoods of 0 for <math>X_{\mathcal{G'}}</math> is formed by the [[Polar set]]s <math>G'^\circ := \{x \in X : \sup_{x' \in G'} |\langle x', x \rangle | \le 1\}</math> as <math>G'</math> varies over <math>\mathcal{G'}</math>.
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"
====The weak topology ''σ''(''X'', ''X''<sup>*</sup>)====
|-
! <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>, ''X'' will have the '''weak topology'' or ''the topology of pointwise convergence on <math>X^*</math>''', which is denoted by <math>\sigma(X, X^*)</math> and ''X'' with this topology is denoted by <math>X_{\sigma}</math> or by <math>X_{\sigma(X, X^*)}</math> if there may be ambiguity.
*By Suppose that ''X'' and ''Y'' are Hausdorff locally convex spaces with ''X'' metrizable and thatletting <math>u : X \to Ymathcal{G}</math> isbe athe linearset map.of all finite subsets Thenof <math>u : X \to Y,</math> is continuous if and only if <math>u : \sigmaL(X, X^*) \to \sigma(Y,; Y^*)</math> iswill continuous.have Thatthe is,'''weak topology on <math>u : L(X \to; Y)</math>''' isor continuous'''the whentopology of pointwise convergence''X'' andor ''Y''the carrytopology theirof givensimple topologies ifconvergence''' and only if <math>uL(X; Y)</math> with this topology is continuousdenoted whenby ''<math>L_{\sigma}(X''; and ''Y'')</math>. carry their weak topologies.
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>.
====Convergence on equicontinuous sets ''ε''(''X'', ''X''<sup>*</sup>)====
The weak-topology on <math>L(X; Y)</math> has the following properties:
By letting <math>\mathcal{G'}</math> be the set of all equicontinuous subsets <math>X^*</math>, ''X'' will have '''the topology of uniform convergence on equicontinuous subsets of <math>X^*</math>''', which is denoted by <math>\epsilon(X, X^*)</math> and ''X'' with this topology is denoted by <math>X_{\epsilon}</math> or by <math>X_{\epsilon(X, X^*)}</math>.
<ul>
* If <math>\mathcal{G'}</math> was the set of all convex balanced weakly compact equicontinuous subsets of <math>X^*</math>, then the same topology would have been induced.
<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}}
* If ''X'' is locally convex and Hausdorff then ''X'''s given topology (i.e. the topology that ''X'' started with) is exactly <math>\epsilon(X, X^*)</math>.
* 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'''
====Mackey topology ''τ''(''X'', ''X''<sup>*</sup>)====
<ul>
By letting <math>\mathcal{G'}</math> be the set of all convex balanced weakly compact subsets of <math>X^*</math>, ''X'' will have the '''Mackey topology on ''X''''' or '''the topology of uniform convergence on convex balanced weakly compact subsets of <math>X^*</math>''', which is denoted by <math>\tau(X, X^*)</math> and ''X'' with this topology is denoted by <math>X_{\tau}</math> or by <math>X_{\tau(X, X^*)}</math>.
<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====
* Suppose that ''X'' is a locally convex Hausdorff space. If ''X'' is metrizable or [[barrled space|barrelled]] then the initial topology of ''X'' is identical to the Mackey topology <math>\tau(X, X^*)</math>.
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>.
====Bounded convergence ''b''(''X'', ''X''<sup>*</sup>)====
The topology of compact convergence on <math>L(X; Y)</math> has the following properties:
By letting <math>\mathcal{G}</math> be the set of all bounded subsets of ''X'', <math>X^*</math> will have '''the topology of bounded convergence''' or '''the topology of uniform convergence on bounded sets''', which is denoted by <math>b(X, X^*)</math> and <math>X^*</math> with this topology is denoted by <math>X^*_{b}</math> or by <math>X^*_{b(X, X^*)}</math>.
<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====
===The Mackey–Arens theorem===
Let ''X'' be a vector space and let ''Y'' be a vector subspace of the algebraic dual of ''X'' that [[Separating set|separates points]] on ''X''. Any locally convex Hausdorff topological vector space (TVS) topology on ''X'' with the property that when ''X'' is equipped with this topology has ''Y'' as its continuous dual space is said to be '''compatible with duality between ''X'' and ''Y'''''. If we give ''X'' the weak topology <math>\sigma(X, Y)</math> then <math>X_{\sigma(X, Y)}</math> is a Hausdorff locally convex topological vector space (TVS) and <math>\sigma(X, Y)</math> is compatible with duality between ''X'' and ''Y'' (i.e. <math>X_{\sigma(X, Y)}^* = (X_{\sigma(X, Y)})^* = Y</math>). We can now ask the question: what are ''all'' of the locally convex Hausdorff TVS topologies that we can place on ''X'' that are compatible with duality between ''X'' and ''Y''? The answer to this question is called the [[Mackey–Arens theorem#Mackey.E2.80.93Arens theorem|Mackey–Arens theorem]]:<ref>Treves, pp. 196, 368 - 370</ref>
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}}
<blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">
'''Theorem.''' Let ''X'' be a vector space and let <math>\mathcal{T}</math> be a locally convex Hausdorff topological vector space topology on ''X''. Let <math>X^*</math> denote the continuous dual space of ''X'' and let <math>X_{\mathcal{T}}</math> denote ''X'' with the topology <math>\mathcal{T}</math>. Then the following are equivalent:
{{ordered list|
| <math>\mathcal{T}</math> is identical to a <math>\mathcal{G'}</math>-topology on ''X'', where <math>\mathcal{G'}</math> is a covering of <math>X^*</math> consisting of convex, balanced, <math>\sigma(X^*, X)</math>-compact sets with the properties that
{{ordered list|type=lower-roman|style=margin-left: 1em
| If <math>G_1', G_2' \in \mathcal{G'}</math> then there exists a <math>G' \in \mathcal{G'}</math> such that <math>G_1' \cup G_2' \subseteq G'</math>, and
| If <math>G_1' \in \mathcal{G'}</math> and <math>\lambda</math> is a scalar then there exists a <math>G' \in \mathcal{G'}</math> such that <math>\lambda G_1' \subseteq G'</math>.
}}
| The continuous dual of <math>X_{\mathcal{T}}</math> is identical to <math>X^*</math>.
}}
And furthermore,
{{ordered list
| the topology <math>\mathcal{T}</math> is identical to the <math>\epsilon(X, X^*)</math> topology, that is, to the topology of uniform on convergence on the equicontinuous subsets of <math>X^*</math>.
| the Mackey topology <math>\tau(X, X^*)</math> is the finest locally convex Hausdorff TVS topology on ''X'' that is compatible with duality between ''X'' and <math>X_{\mathcal{T}}^*</math>, and
| the weak topology <math>\sigma(X, X^*)</math> is the weakest locally convex Hausdorff TVS topology on ''X'' that is compatible with duality between ''X'' and <math>X_{\mathcal{T}}^*</math>.
}}
</blockquote>
The topology of bounded convergence on <math>L(X; Y)</math> has the following properties:
==G-H-topologies on spaces of bilinear maps==
<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==
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 its subspace 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 ''L(X, Y)'' we can place a topology on <math>\mathcal{B}(X, Y; Z)</math> and <math>B(X, Y; Z)</math>.
{{Main|Polar topology}}
Throughout, we assume that <math>X</math> is a TVS.
Let <math>\mathcal{G}</math> be a set of subsets of ''X'', <math>\mathcal{H}</math> be a set of subsets of ''Y''. Let <math>\mathcal{G} \times \mathcal{H}</math> denote the collection of all sets ''G'' × ''H'' 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>'''.
===𝒢-topologies versus polar topologies===
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 ''X''. 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}</math>-<math>\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:
* ''X'' and ''Y'' are barrelled spaces and <math>Z</math> is locally convex.
* ''X'' is a [[F-space]], ''Y'' 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.
* ''X'' is normed and ''Y'' and <math>Z</math> the strong duals of reflexive Fréchet spaces.
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.
===The ''ε''-topology===
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
SupposeHowever, thatif <math>X, Y</math>, andis <math>Z</math>a TVS whose bounded subsets are locally{{em|not}} convexexactly spacesthe andsame letas its <math>\mathcal{G{em|weakly}</math>'} bounded subsets, then the notion of "bounded andin <math>\mathcal{H}X</math>'" beis thestronger collectionsthan ofthe equicontinuous subsetsnotion of "<math>\sigma\left(X, X^*{\prime}\right)</math>-bounded andin <math>Y^*X</math>," respectively(i.e. Thenbounded thein <math>\mathcal{G}X</math>'- implies <math>\mathcalsigma\left(X, X^{H\prime}\right)</math>'-topologybounded onin <math>\mathcal{B}(X^*_{b(X^*, X)}, Y^*_{b(X^*, X)}; Z)</math>) willso bethat a topological vector space topology. This topology is called the ε-topology and <math>\mathcal{BG}(X^*_{b(X^*, X)}, Y_{b(X^*, X)}; Z)</math> with this -topology it is denoted byon <math>\mathcal{B}_{\epsilon}(X^*_{b(X^*, X)\prime}, Y^*_{b(X^*, X)}; Z)</math> or(as simplydefined byin this article) is <math>\mathcal{B}_{\epsilonem|not}(X^*_{b}, Y^*_{b};necessarily Z)</math>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]].
Part of the importance of this vector space and this topology is that it contains many subspace, such as <math>\mathcal{B}(X^*_{\sigma(X^*, X)}, Y^*_{\sigma(X^*, X)}; Z)</math>, which we denote by <math>\mathcal{B}(X^*_{\sigma}, Y^*_{\sigma}; Z)</math>. When this subspace is given the subspace topology of <math>\mathcal{B}_{\epsilon}</math><math>(X^*_{b}, Y^*_{b}; Z)</math> it is denoted by <math>\mathcal{B}_{\epsilon}(X^*_{\sigma}, Y^*_{\sigma}; Z)</math>.
We list here some of the most common polar topologies.
===List of polar topologies===
In the instance where ''Z'' is the field of these vector spaces <math>\mathcal{B}(X^*_{\sigma}, Y^*_{\sigma})</math> is a [[tensor product]] of ''X'' and ''Y''. In fact, if ''X'' and ''Y'' are locally convex Hausdorff spaces then <math>\mathcal{B}(X^*_{\sigma}, Y^*_{\sigma})</math> is vector space isomorphic to <math>L(X^*_{\sigma(X^*, X)}, Y_{\sigma(Y^*, Y)})</math>, which is in turn equal to <math>L(X^*_{\tau(X^*, X)}, Y)</math>.
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}_{\epsilonvarepsilon}</math><math>\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 (orrespectively, both Banach) then so is <math>\mathcal{B}_{\epsilon}</math><math>\left(X^*{\prime}_{\sigma}, Y^*{\prime}_{\sigma}\right)</math>
== See also==
* [[Bornological space]]
* [[Bounded linear operator]]
* [[Operator norm]]
* [[Uniform convergence]]
* [[Uniform space]]
* [[Polar topology]]
* {{annotated link|Bornological space}}
==Notes==
* {{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}}
== References Bibliography==
* {{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}}
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | year=1970 | isbn=0-387-05380-8 | pages=61–63 }}
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn|Hogbe-Nlend|1977|p=}} -->
* {{Cite book | isbn = 9780486453521 | title = Topological Vector Spaces, Distributions and Kernels | last1 = Trèves | first1 = François | year = 1995 | publisher = [[Dover Publications]] | pages = 136–149, 195–201, 240–252, 335–390, 420–433 }}
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Cite book | isbn = 9783540115656 | title = Counterexamples in Topological Vector Spaces | last1 = Khaleelulla | first1 = S.M. | year = 1982 | publisher = [[Springer-Verlag]] | ___location = Berlin Heidelberg | series = [[Graduate Texts in Mathematics|GTM]] | volume = 936 | pages = 29–33, 49, 104 }}
* {{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=}} -->
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{{Duality and spaces of linear maps}}
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