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Line 1: 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. ~~{{no footnotes\|date=January 2014}}~~ In [[mathematics]], a '''[[linear map]]''' (also called a '''linear mapping''', '''linear [[Transformation (function)\|transformation]]''' or, in some contexts, '''[[linear function]]''') is a [[function (mathematics)\|mapping]] {{math\|''V'' ↦ ''W''}} between two [[Module (mathematics)\|module]]s (including [[vector space]]s) that preserves the operations of addition and [[scalar (mathematics)\|scalar]] multiplication. 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\|bornolgies]], then one can study the subspace of linear maps that preserve this structure. ==Topologies of uniform convergence on arbitrary spaces of maps== ~~=Spaces of Continuous Linear Maps=~~ Throughout, the following is assumed: Throughout this section we will assume that <math>X</math> and <math>Y</math> are [[topological vector space]]s and we will let <math>L(X, Y)</math>, denote the vector space of all continuous linear maps from <math>X</math> and <math>Y</math>. <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> ==G=𝒢-~~topologies~~topology=== The following sets will constitute the basic open subsets of topologies on spaces of linear maps. We can form topologies on <math>L(X, Y)</math> in the following way. Let <math>\mathcal{G}</math> be a set of subsets of <math>X</math> and let <math>\mathcal{N}</math> be a basis of neighborhoods of 0 in <math>Y</math>. Then we can define a topology on <math>L(X, Y)</math> by defining the neighborhoods of 0 in <math>L_{\mathcal{G}}(X, Y)</math> to be :For any subsets <math>~~\mathcal{U}(~~G~~, V) =~~ \{fsubseteq ~~\in~~T</math> ~~L(X,~~and ~~Y) : f(G)~~<math>N \~~sub~~subseteq ~~N\}~~Y,</math> let <math display="block">\mathcal{U}(G, N) := \{f \in F : f(G) \subseteq N\}.</math> as ''G'' and ''N'' range over all <math>G \in \mathcal{G}</math> and <math>N \in \mathcal{N}</math>. This topology is known as the '''<math>\mathcal{G}</math>-topology on <math>L(X, Y)</math>''' or as the '''topology of uniform convergence on the sets in <math>\mathcal{G}</math>''' and <math>L(X, Y)</math> with this topology is and denoted by <math>L_{\mathcal{G}}(X, Y)</math>. The family 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> (that is, <math>L_{\mathcal{G}}(X, Y)</math> is a topological vector space) 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>; we will assume this to be the case for the rest of the article. Note that in particular, this is the case if <math>\mathcal{G}</math> consists of (von-Neumann) bounded subsets of <math>X</math>. <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> ~~Often, <math>\mathcal{G}</math> is required to satisfy the following two axioms:~~ 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}} \|- ~~\| 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>. \|} IfCall a subset <math>~~\mathcal{G}~~B</math> isof a<math>T</math> ~~[[bornology]]~~'''<math>F</math>-bounded''' onif <math>Xf(B)</math>~~. which~~ is ~~often~~a ~~the~~bounded ~~case,~~subset ~~then~~of ~~these~~<math>Y</math> ~~two~~for ~~axioms~~every ~~are~~<math>f ~~satisfied~~\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''' ~~==Examples==~~ 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> ~~===The topology of pointwise convergence <math>L_{\sigma}(X, Y)</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 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''' and <math>L_(X, Y)</math> with this topology is denoted by <math>L_{\sigma}(X, Y)</math> <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}} ~~The weak-topology on <math>L(X, Y)</math> has the following properties:~~ <math display=block>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math> * The weak-closure of an equicontinuous subset of <math>L(X, Y)</math> is equicontinuous. which implies: * If <math>Y</math> is locally convex, then the convex balanced hull of an equicontinuous subset of <math>L_(X, Y)</math> is equicontinuous. <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> 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> ~~===Compact-Convex Convergence <math>L_{\gamma}(X, Y)</math>===~~ ===Uniform structure=== By letting <math>\mathcal{G}</math> be the set of all compact convex subsets of <math>X</math>, <math>L(X, Y)</math> will have the ''the topology of compact convex convergence''' or '''the topology of uniform convergence on compact convex sets'''<math>L(X, Y)</math> with this topology is denoted by <math>L_{\gamma}(X, Y)</math>. {{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 ~~===Compact Convergence <math>L_{c}(X, Y)</math>>===~~ <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''' 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 ''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>. 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}} * If <math>X</math> is a [[Frechet space]] or a [[LF-space]] and <math>Y</math> is a complete locally convex Hausdorff space then <math>L_{c}(X, Y)</math> is complete. {{Math theorem\|name=Theorem{{sfn\|Jarchow\|1981\|pp=43-55}}\|math_statement= ~~===Strong dual topology <math>L_{b}(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=== 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 ''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>. '''Local convexity''' ~~The topology of bounded convergence on <math>L(X, Y)</math> has the following properties:~~ * If <math>X</math> and <math>Y</math> are normed spaces then so is <math>L_{b}(X, Y)</math> * If <math>X</math> and <math>Y</math> are normed spaces and <math>u : X \to Y</math> is a continuous linear map, then the norm of <math>u</math> and the norm of the [[transpose]] of <math>u</math> are equal.<ref>Treves pp. 242 - 243</ref> 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: ~~===Properties===~~ <math display="block">p_{G,i}(f) := \sup_{x \in G} p_i(f(x)),</math> * If <math>Y</math> is locally convex then so is <math>L_{\mathcal{G}}(X, Y)</math>. * Ifas <math>YG</math> isvaries ~~Hausdorff and~~over <math>~~\bigcup_{G \in~~ \mathcal{G}~~} G~~</math> ~~is dense in~~and <math>Xi</math> ~~then~~varies over <math>~~L_{\mathcal{G}}(X, Y)~~I</math> ~~is Hausdorff~~.{{sfn\|Schaefer\|Wolff\|1999\|p=81}} * If <math>X</math> is a [[bornological space]] and if <math>Y</math> is a [[Complete_metric_space#Topologically_complete_spaces\|complete]] locally convex Hausdorff space then <math>L_{b}(X, Y)</math> is complete. * If <math>X</math> is a [[Frechet space]] or a [[LF-space]] and if <math>Y</math> is a [[Complete_metric_space#Topologically_complete_spaces\|complete]] locally convex Hausdorff space then <math>L_{c}(X, Y)</math> is complete. * If <math>X</math> and <math>Y</math> are both normed spaces then <math>L_{b}(X, Y)</math> is a normed space with the usual operator norm. '''Hausdorffness''' ~~=G-topologies on the continuous dual induced by ''X''=~~ 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}} The [[Dual space#Continuous dual space\|continuous dual space]] of a topological vector space <math>X</math> 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 <math>X</math>, 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: A basis of neighborhoods of ''0'' for <math>X^_{\mathcal{G}}</math> is formed, as as <math>G</math> varies over <math>\mathcal{G}</math>, by the [[polar set]]s <math>G^\circ := \{x' \in X^ : \sup_{x \in G} \|\langle x', x \rangle \| \le 1\}</math>. ** 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 <math>G \sub X</math> is bounded then <math>G^\circ</math> is absorbing, so <math>\mathcal{G}</math> usually consists of bounded subsets of <math>X</math>. * <math>X^_{\mathcal{G}}</math> is locally convex, If <math>\bigcup_{G \in \mathcal{G}} G</math> is dense in <math>X</math> then <math>X^_{\mathcal{G}}</math> is Hausdorff. If <math>\bigcup_{G \in \mathcal{G}} G</math> covers <math>X</math> then the canonical map from <math>X</math> 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 <math>X</math> then the canonical map of <math>X</math> into <math>(X^_{\mathcal{G}})^</math> is an injection. Suppose that <math>X</math> and <math>Y</math> 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 <math>X</math> and <math>Y</math>, 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 <math>u(G) \subseteq H</math>.<ref>Treves pp. 199 - 200</ref> ** In particular, the transpose of <math>u</math> is continuous if <math>X^</math> caries 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 <math>X</math> 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 <math>X</math> 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</math> is continuous if and only if <math>X</math> 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 <math>T</math> is a topological space. ~~==Examples==~~ 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. '''Boundedness''' ~~===The Weak topology <math>\sigma(X^, X)</math> or the Weak* topology===~~ ByA ~~letting~~subset <math>~~\mathcal{G}~~H</math> ~~be the set of all finite subsets~~ of <math>XF</math>, ~~<math>X^</math>~~is ~~will~~[[Bounded ~~have~~set ~~the~~(topological ~~'''weak~~vector ~~topology~~space)\|bounded]] onin the <math>X^\mathcal{G}</math>''' more comonly as the '''weak* -topology~~'''~~ orif ~~''the~~and ~~topology~~only ~~of pointwise convergence''', which is~~if ~~denoted~~for byevery <math>G \~~sigma(X^,~~in X)\mathcal{G},</math> ~~and~~ <math>~~X^</math>~~H(G) ~~with~~= ~~this~~\bigcup_{h ~~topology~~\in isH} ~~denoted by <math>X^_{\sigma}~~h(G)</math> oris bybounded in <math>~~X^_{\sigma(X^, X)}~~Y.</math> ~~if there may be ambiguity.~~{{sfn\|Schaefer\|Wolff\|1999\|p=81}} ===Examples of 𝒢-topologies=== ~~The <math>\sigma(X^, X)</math> topology has the following properties:~~ * <math>X^</math> with the <math>\sigma(X^, X)</math> is normable if and only if <math>X</math> is finite dimensional. * When <math>X</math> 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. '''Pointwise convergence''' ~~===Compact-Convex Convergence <math>\gamma(X^, X)</math>===~~ ByIf ~~letting~~we let <math>\mathcal{G}</math> be the set of all ~~compact convex~~finite subsets of <math>XT</math>, ~~<math>X^</math> will have~~then the ~~''the topology of compact convex convergence''' or '''the topology of uniform convergence on compact convex sets''', which is denoted by~~ <math>\~~gamma(X^, X)~~mathcal{G}</math>-topology ~~and~~on <math>X^F</math> ~~with this topology~~ is ~~denoted~~called bythe ~~<math>X^_{\gamma}</math>~~'''topology orof bypointwise ~~<math>X^_{\gamma(X^,~~convergence'''. ~~X)}</math>.~~ 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}} ~~===Compact Convergence <math>c(X^, X)</math>===~~ ==𝒢-topologies on spaces of continuous linear maps== By letting <math>\mathcal{G}</math> be the set of all compact subsets of <math>X</math>, <math>X^</math> will have the ''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>. Throughout Ifthis section we will assume that <math>X</math> ~~is a [[Frechet space]] or a [[LF-space]] then~~and <math>~~L_{c}(X,~~ Y)</math> isare ~~complete~~[[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. ~~===Mackey topology <math>\tau(X^, X)</math>===~~ 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=== By letting <math>\mathcal{G}</math> be the set of all convex balanced weakly compact subsets of <math>X</math>, <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>. '''Assumptions that guarantee a vector topology''' ~~===Strong dual topology <math>b(X^, X)</math>===~~ ~~By letting~~ (<math>\mathcal{G}</math> beis ~~the set of all bounded subsets of~~directed): <math>X</math>, <math>X^\mathcal{G}</math> will ~~have~~be ~~the~~a ~~''the~~non-empty ~~topology~~collection of ~~bounded convergence on <math>X</math>''' or '''the topology~~subsets of ~~uniform convergence on bounded sets''' or the '''strong dual topology on~~ <math>X^</math>~~''',~~ ~~which~~[[Directed ~~is denoted~~set\|directed]] by ~~<math>b~~(~~X^, X~~subset)~~</math>~~ ~~and~~inclusion. ~~<math>X^</math> with this topology~~That is, ~~denoted~~for byany <math>~~X^_{b}</math>~~G, orH by\in ~~<math>X^_~~\mathcal{b(X^G}, ~~X)}~~</math>. ~~Due~~there ~~to its importance, the continuous dual space of~~exists <math>~~X^_{b}</math>,~~K ~~which~~\in ~~is commonly denoted by <math>X^~~\mathcal{*G}</math> sosuch that <math>(X^_{b})^G =\cup ~~X^{}~~H \subseteq K</math>. The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]]. ~~The <math>b(X^, X)</math> topology has the following properties:~~ The next assumption will guarantee that the sets <math>\mathcal{U}(G, N)</math> are [[Balanced set\|balanced]]. * If <math>X</math> 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 <math>X</math>. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome. If <math>X</math> is a [[bornological space]] (ex: [[metrizable]] or [[LF-space]]) then <math>X^_{b(X^, X)}</math>is complete. * If <math>X</math> is a normed space then the stong 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> * (<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. * Given a Hausdorff locally convex topological vector space <math>X</math>, the following properties are equivalent:<ref>Treves, p. 201</ref> ~~{{ordered list\|type=lower-roman\|style=margin-left: 1em~~ 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> ~~\| <math>X^_{b(X^, X)}</math> is metrizable,~~ ~~\| <math>X^_{b(X^, X)}</math> is normable,~~ * (<math>G \in \mathcal{G}</math> are bounded): <math>\mathcal{G}</math> is assumed to consist entirely of bounded subsets of <math>X.</math> ~~\| <math>X</math> is normable.~~ 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> }} * If <math>X</math> 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 [[Frechet space]] if and only if all <math>X_k</math> are normable. '''Common assumptions''' ~~===Mackey topology <math>\tau(X^, X^{})</math>===~~ 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: 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>. :<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>). This topology is finer than <math>b(X^, X)</math> and hence finer than <math>\tau(X^, X)</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: ~~===Other examples===~~ :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=== ~~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''. '''Hausdorffness''' ~~=G-topologies on ''X'' induced by the continuous dual=~~ 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> There is a canonical map from <math>X</math> 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 <math>X</math> as being contained in the continuous dual of <math>X^_{\sigma}</math> (that is, contined 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 <math>X</math> 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. 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}} ~~Since we are now regarding~~If <math>XF</math> asis the ~~continuous dual~~vector ~~space~~subspace of <math>XY^_{\sigma}T</math>, weconsisting ~~can~~of ~~look~~all atcontinuous ~~sets~~linear ofmaps ~~subsets~~that ofare bounded on every <math>~~X^_~~G \in \mathcal{~~\sigma~~G},</math>, ~~say~~then the <math>\mathcal{G'}</math> ~~and construct a dual space~~ -topology on ~~the dual of~~ <math>~~X^_{\sigma}~~F</math>~~, which~~ is Hausdorff if <math>XY</math>. is AHausdorff ~~basis of neighborhoods of ''0'' for~~and <math>~~X_{~~\mathcal{G'}}</math> is ~~formed~~total ~~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'~~T.</math> ~~varies over <math>\mathcal~~{G'{sfn\|Narici\|Beckenstein\|2011\|pp=371-423}}~~</math>.~~ '''Completeness''' ~~==Examples==~~ 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> ~~===The Weak topology <math>\sigma(X, X^)</math>===~~ <ul> By letting <math>\mathcal{G'}</math> be the set of all finite subsets of <math>X'</math>, <math>X</math> 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 <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. <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''' ~~===Convergence on Equicontinuous sets <math>\epsilon(X, X^)</math>===~~ 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> By letting <math>\mathcal{G'}</math> be the set of all equicontinuous subsets <math>X^</math>, <math>X</math> will have the '''the topology of uniform convergence on equicontinuous subsets of <math>X^</math>''', which is denoted by <math>\epsilon(X, X^)</math> and <math>X</math> with this topology is denoted by <math>X_{\epsilon}</math> or by <math>X_{\epsilon(X, X^)}</math>. Then the following are equivalent:{{sfn\|Schaefer\|Wolff\|1999\|p=81}} 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. <ol> If <math>X</math> is locally convex and Hausdorff then <math>X</math>'s given topology (i.e. the topology that <math>X</math> started with) is exactly <math>\epsilon(X, X^)</math>. <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}} ~~===Mackey topology <math>\tau(X, X^)</math>===~~ 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=== By letting <math>\mathcal{G'}</math> be the set of all convex balanced weakly compact subsets of <math>X^</math>, <math>X</math> will have the '''Mackey topology on <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}</math> or by <math>X_{\tau(X, X^)}</math>. {\| class="wikitable" ~~===Bounded Convergence <math>b(X, X^)</math>===~~ \|- ! <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 bounded subsets of <math>X</math>, <math>X^</math> will have the ''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>. 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 Mackey-Arens Theorem==~~ Let <math>X</math> be a vector space and let <math>Y</math> be a vector subspace of the algebraic dual of <math>X</math> that [[Separating set\|separates points]] on <math>X</math>. Any locally convex Hausdorff topological vector space (TVS) topology on <math>X</math> with the property that when <math>X</math> is equipped with this topology has <math>Y</math> as its the continuous dual space is said to be '''compatible with duality between <math>X</math> and <math>Y</math>'''. If we give <math>X</math> 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 <math>X</math> and <math>Y</math> (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 <math>X</math> that are compatible with duality between <math>X</math> and <math>Y</math>? 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> The weak-topology on <math>L(X; Y)</math> has the following properties: ~~<blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">~~ <ul> '''Theorem.''' Let <math>X</math> be a vector space and let <math>\mathcal{T}</math> be a locally convex Hausdorff topological vector space topology on <math>X</math>. Let <math>X^</math> denote the continuous dual space of <math>X</math> and let <math>X_{\mathcal{T}}</math> denote <math>X</math> with the topology <math>\mathcal{T}</math>. Then the following are equivalent: <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}} ~~{{ordered list\|~~ So in particular, on every equicontinuous subset of <math>L(X; Y),</math> the topology of pointwise convergence is metrizable.</li> \| <math>\mathcal{T}</math> is identical to a <math>\mathcal{G'}</math>-topology on <math>X</math>, 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 <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>. ~~{{ordered list\|type=lower-roman\|style=margin-left: 1em~~ * In ~~\| If~~addition, <math>~~G_1',~~L(X; ~~G_2' \in \mathcal{G'}~~Y)</math> ~~then~~is ~~there~~dense ~~exists~~in the aspace of all linear maps (continuous or not) <math>~~G' \in \mathcal{G'}~~X</math> ~~such that~~into <math>~~G_1' \cup G_2' \subseteq G'~~Y.</math>~~, and~~</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> ~~\| 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>.~~ </ul> }} ~~\| 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 <math>X</math> that is compatible with duality between <math>X</math> and <math>X_{\mathcal{T}}^</math>, and \| the weak topology <math>\sigma(X, X^)</math> is the weakest locally convex Hausdorff TVS topology on <math>X</math> that is compatible with duality between <math>X</math> and <math>X_{\mathcal{T}}^</math>. }} '''Equicontinuous subsets''' <ul> ~~</blockquote>~~ <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==== ~~=G-H-topologies on Spaces of Bilinear maps=~~ WeBy ~~will let~~letting <math>\mathcal{BG}~~(X, Y; Z)~~</math> ~~denote~~be the ~~space~~set of ~~separately~~all ~~continuous~~compact ~~bilinear~~subsets ~~maps and~~of <math>B(X, ~~Y; Z)~~</math> ~~denote its subspace the space of continuous bilinear maps, where~~ <math>L(X,; Y)</math> ~~and <math>Z</math> are topological vector space~~will ~~over~~have '''the ~~same~~topology ~~field~~of ~~(either~~compact ~~the real~~convergence''' or ~~complex~~'''the ~~numbers).~~topology Inof anuniform ~~analogous~~convergence ~~way~~on tocompact ~~how~~sets''' ~~we placed a topology on~~and <math>L(X,; Y)</math> wewith ~~can~~this ~~place~~topology ais ~~topology~~denoted onby <math>~~\mathcal{B}~~L_c(X~~, Y~~; ~~Z)</math> and <math>B(X,~~ Y~~; Z~~)</math>. The topology of compact convergence on <math>L(X; Y)</math> has the following properties: Let <math>\mathcal{G}</math> be a set of subsets of <math>X</math>, <math>\mathcal{H}</math> be a set of subsets of <math>Y</math> and let <math>\mathcal{N}</math> be a basis of neighborhoods of 0 in <math>Z</math>. Then we can define a topology on <math>\mathcal{B}(X, Y; Z)</math> by defining the neighborhoods of 0 in <math>\mathcal{B}(X, Y; Z)</math> to be <ul> ~~:<math>\mathcal{U}(G, H, V) = \{b \in \mathcal{B}(X, Y; Z) : f(G, H) \sub N\}</math>~~ <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> as ''G'', ''H'', and ''N'' range over all <math>G \in \mathcal{G}</math>, <math>H \in \mathcal{H}</math> and <math>N \in \mathcal{N}</math>. This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math>-topology on <math>\mathcal{B}(X, Y; Z)</math>''' or as the '''topology of uniform convergence on the products <math>G \times H</math> of <math>\mathcal{G} \times \mathcal{H}</math>'''. We can now give <math>B(X, Y; Z)</math> the subspace topology inherited from <math>\mathcal{B}(X, Y; Z)</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 additional requirements. <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: ~~= See also=~~ <ul> * [[Bornological space]] <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> * [[Bounded linear operator]] <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}} * [[Operator norm]] * 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== ~~= References =~~ {{Main\|Polar topology}} * {{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}} * {{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 }} Throughout, we assume that <math>X</math> is a TVS. * {{cite book \| last = Tr\`{e}ves \| first = Fran\c{c}ois \| title=Topological Vector Spaces, Distributions and Kernels \| publisher=[[Academic Press, Inc.]] \| year=1995 \| isbn=0-486-45352-9 \| pages=136-149, 195-201, 240-252, 335-390, 420-433 }} * {{cite book \| author=S.M. Khaleelulla \| title=Counterexamples in Topological Vector Spaces \| publisher=[[Springer-Verlag]] \| series=[[Graduate Texts in Mathematics\|GTM]] \| volume=936 \| date=1982 \| isbn=978-3-540-11565-6 \| pages=29-33, 49, 104}} ===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== ~~{{Functional Analysis}}~~ * {{Grothendieck Topological Vector Spaces}} <!--{{sfn\|Grothendieck\|1973\|p=}}--> ~~[[Category:Topological vector spaces]]~~ * {{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]] ~~{{mathanalysis-stub}}~~ [[Category:Topological vector spaces]] [[Category:Topology of function spaces]]