is a vector subspace of [note 1] which denotes the set of all -valued functions with domain
𝒢-topologyedit
The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets and let
The family
forms a neighborhood basis[1] at the origin for a unique translation-invariant topology on where this topology is not necessarily a vector topology (that is, it might not make into a TVS). This topology does not depend on the neighborhood basis that was chosen and it is known as the topology of uniform convergence on the sets in or as the -topology.[2] However, this name is frequently changed according to the types of sets that make up (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details[3]).
A subset of is said to be fundamental with respect to if each is a subset of some element in In this case, the collection can be replaced by without changing the topology on [2] One may also replace with the collection of all subsets of all finite unions of elements of without changing the resulting -topology on [4]
Call a subset of -bounded if is a bounded subset of for every [5]
Theorem[2][5] — The -topology on is compatible with the vector space structure of if and only if every is -bounded; that is, if and only if for every and every is bounded in
Properties
Properties of the basic open sets will now be described, so assume that and Then is an absorbing subset of if and only if for all absorbs .[6] If is balanced[6] (respectively, convex) then so is
The equality always holds. If is a scalar then so that in particular, [6] Moreover,[4]
Given the family of all sets as ranges over any fundamental system of entourages of forms a fundamental system of entourages for a uniform structure on called the uniformity of uniform converges on or simply the -convergence uniform structure.[7] The -convergence uniform structure is the least upper bound of all -convergence uniform structures as ranges over [7]
Nets and uniform convergence
Let and let be a net in Then for any subset of say that converges uniformly to on if for every there exists some such that for every satisfying (or equivalently, for every ).[5]
Theorem[5] — If and if is a net in then in the -topology on if and only if for every converges uniformly to on
Inherited propertiesedit
Local convexity
If is locally convex then so is the -topology on and if is a family of continuous seminorms generating this topology on then the -topology is induced by the following family of seminorms:
If is Hausdorff and then the -topology on is Hausdorff.[5]
Suppose that is a topological space. If is Hausdorff and is the vector subspace of consisting of all continuous maps that are bounded on every and if is dense in then the -topology on is Hausdorff.
Boundedness
A subset of is bounded in the -topology if and only if for every is bounded in [8]
Examples of 𝒢-topologiesedit
Pointwise convergence
If we let be the set of all finite subsets of then the -topology on is called the topology of pointwise convergence. The topology of pointwise convergence on is identical to the subspace topology that inherits from when is endowed with the usual product topology.
If is a non-trivial completely regular Hausdorff topological space and is the space of all real (or complex) valued continuous functions on the topology of pointwise convergence on is metrizable if and only if is countable.[5]
𝒢-topologies on spaces of continuous linear mapsedit
Throughout this section we will assume that and are topological vector spaces. will be a non-empty collection of subsets of directed by inclusion. will denote the vector space of all continuous linear maps from into If is given the -topology inherited from then this space with this topology is denoted by . The continuous dual space of a topological vector space over the field (which we will assume to be real or complex numbers) is the vector space and is denoted by .
The -topology on is compatible with the vector space structure of if and only if for all and all the set is bounded in which we will assume to be the case for the rest of the article. Note in particular that this is the case if consists of (von-Neumann) bounded subsets of
Assumptions on 𝒢edit
Assumptions that guarantee a vector topology
( is directed): will be a non-empty collection of subsets of directed by (subset) inclusion. That is, for any there exists such that .
The above assumption guarantees that the collection of sets forms a filter base. The next assumption will guarantee that the sets are balanced. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
( are balanced): is a neighborhoods basis of the origin in that consists entirely of balanced sets.
The following assumption is very commonly made because it will guarantee that each set is absorbing in
( are bounded): is assumed to consist entirely of bounded subsets of
The next theorem gives ways in which can be modified without changing the resulting -topology on
Theorem[6] — Let be a non-empty collection of bounded subsets of Then the -topology on is not altered if is replaced by any of the following collections of (also bounded) subsets of :
Some authors (e.g. Narici) require that satisfy the following condition, which implies, in particular, that is directed by subset inclusion:
is assumed to be closed with respect to the formation of subsets of finite unions of sets in (i.e. every subset of every finite union of sets in belongs to ).
Some authors (e.g. Trèves [9]) require that be directed under subset inclusion and that it satisfy the following condition:
If and is a scalar then there exists a such that
If is a bornology on which is often the case, then these axioms are satisfied. If is a saturated family of bounded subsets of then these axioms are also satisfied.
If is the vector subspace of consisting of all continuous linear maps that are bounded on every then the -topology on is Hausdorff if is Hausdorff and is total in [6]
Completeness
For the following theorems, suppose that is a topological vector space and is a locally convex Hausdorff spaces and is a collection of bounded subsets of that covers is directed by subset inclusion, and satisfies the following condition: if and is a scalar then there exists a such that
Let be a bornological space, a locally convex space, and a family of bounded subsets of such that the range of every null sequence in is contained in some If is quasi-complete (respectively, complete) then so is .[12]
Boundedness
Let and be topological vector spaces and be a subset of Then the following are equivalent:[8]
For every neighborhood of the origin in the set absorbs every
If is a collection of bounded subsets of whose union is total in then every equicontinuous subset of is bounded in the -topology.[11] Furthermore, if and are locally convex Hausdorff spaces then
if is bounded in (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 [13]
if is quasi-complete (meaning that closed and bounded subsets are complete), then the bounded subsets of are identical for all -topologies where is any family of bounded subsets of covering [13]
Examplesedit
("topology of uniform convergence on ...")
Notation
Name ("topology of...")
Alternative name
finite subsets of
pointwise/simple convergence
topology of simple convergence
precompact subsets of
precompact convergence
compact convex subsets of
compact convex convergence
compact subsets of
compact convergence
bounded subsets of
bounded convergence
strong topology
The topology of pointwise convergenceedit
By letting be the set of all finite subsets of will have the weak topology on or the topology of pointwise convergence or the topology of simple convergence and with this topology is denoted by . Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity;[6] for this reason, this article will avoid referring to this topology by this name.
A subset of is called simply bounded or weakly bounded if it is bounded in .
The weak-topology on has the following properties:
If is separable (that is, it has a countable dense subset) and if is a metrizable topological vector space then every equicontinuous subset of is metrizable; if in addition is separable then so is [14]
So in particular, on every equicontinuous subset of the topology of pointwise convergence is metrizable.
Let denote the space of all functions from into If is given the topology of pointwise convergence then space of all linear maps (continuous or not) into is closed in .
In addition, is dense in the space of all linear maps (continuous or not) into
Suppose and are locally convex. Any simply bounded subset of is bounded when has the topology of uniform convergence on convex, balanced, bounded, complete subsets of If in addition is quasi-complete then the families of bounded subsets of are identical for all -topologies on such that is a family of bounded sets covering [13]
If is locally convex, then the convex balanced hull of an equicontinuous subset of is equicontinuous.
Let and be TVSs and assume that (1) is barreled, or else (2) is a Baire space and and are locally convex. Then every simply bounded subset of is equicontinuous.[11]
On an equicontinuous subset of the following topologies are identical: (1) topology of pointwise convergence on a total subset of ; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.[11]
Compact convergenceedit
By letting be the set of all compact subsets of will have the topology of compact convergence or the topology of uniform convergence on compact sets and with this topology is denoted by .
The topology of compact convergence on has the following properties:
The topology of pointwise convergence on a dense subset of
The topology of pointwise convergence on
The topology of compact convergence.
The topology of precompact convergence.
If is a Montel space and is a topological vector space, then and have identical topologies.
Topology of bounded convergenceedit
By letting be the set of all bounded subsets of will have the topology of bounded convergence on or the topology of uniform convergence on bounded sets and with this topology is denoted by .[6]
The topology of bounded convergence on has the following properties:
If is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if is a Hausdorff locally convex space), then a -topology on (as defined in this article) is a polar topology and conversely, every polar topology if a -topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in " is stronger than the notion of "-bounded in " (i.e. bounded in implies -bounded in ) so that a -topology on (as defined in this article) is not necessarily a polar topology. One important difference is that polar topologies are always locally convex while -topologies need not be.
Polar topologies have stronger results than the more general topologies of uniform convergence described
February 27, 2024
topologies, spaces, linear, maps, mathematics, particularly, functional, analysis, spaces, linear, maps, between, vector, spaces, endowed, with, variety, topologies, studying, space, linear, maps, these, topologies, give, insight, into, spaces, themselves, art. In mathematics particularly functional analysis spaces of linear maps between two vector spaces can be endowed with a variety of topologies Studying space of linear maps and these topologies can give insight into the spaces themselves The article operator topologies discusses topologies on spaces of linear maps between normed spaces whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces TVSs Contents 1 Topologies of uniform convergence on arbitrary spaces of maps 1 1 𝒢 topology 1 2 Uniform structure 1 3 Inherited properties 1 4 Examples of 𝒢 topologies 2 𝒢 topologies on spaces of continuous linear maps 2 1 Assumptions on 𝒢 2 2 Properties 2 3 Examples 2 3 1 The topology of pointwise convergence 2 3 2 Compact convergence 2 3 3 Topology of bounded convergence 3 Polar topologies 3 1 𝒢 topologies versus polar topologies 3 2 List of polar topologies 4 𝒢 ℋ topologies on spaces of bilinear maps 4 1 The e topology 5 See also 6 References 7 BibliographyTopologies of uniform convergence on arbitrary spaces of maps editThroughout the following is assumed T displaystyle T nbsp is any non empty set and G displaystyle mathcal G nbsp is a non empty collection of subsets of T displaystyle T nbsp directed by subset inclusion i e for any G H G displaystyle G H in mathcal G nbsp there exists some K G displaystyle K in mathcal G nbsp such that G H K displaystyle G cup H subseteq K nbsp Y displaystyle Y nbsp is a topological vector space not necessarily Hausdorff or locally convex N displaystyle mathcal N nbsp is a basis of neighborhoods of 0 in Y displaystyle Y nbsp F displaystyle F nbsp is a vector subspace of Y T t T Y displaystyle Y T prod t in T Y nbsp note 1 which denotes the set of all Y displaystyle Y nbsp valued functions f T Y displaystyle f T to Y nbsp with domain T displaystyle T nbsp 𝒢 topology edit The following sets will constitute the basic open subsets of topologies on spaces of linear maps For any subsets G T displaystyle G subseteq T nbsp and N Y displaystyle N subseteq Y nbsp letU G N f F f G N displaystyle mathcal U G N f in F f G subseteq N nbsp The family U G N G G N N displaystyle mathcal U G N G in mathcal G N in mathcal N nbsp forms a neighborhood basis 1 at the origin for a unique translation invariant topology on F displaystyle F nbsp where this topology is not necessarily a vector topology that is it might not make F displaystyle F nbsp into a TVS This topology does not depend on the neighborhood basis N displaystyle mathcal N nbsp that was chosen and it is known as the topology of uniform convergence on the sets in G displaystyle mathcal G nbsp or as the G displaystyle mathcal G nbsp topology 2 However this name is frequently changed according to the types of sets that make up G displaystyle mathcal G nbsp e g the topology of uniform convergence on compact sets or the topology of compact convergence see the footnote for more details 3 A subset G 1 displaystyle mathcal G 1 nbsp of G displaystyle mathcal G nbsp is said to be fundamental with respect to G displaystyle mathcal G nbsp if each G G displaystyle G in mathcal G nbsp is a subset of some element in G 1 displaystyle mathcal G 1 nbsp In this case the collection G displaystyle mathcal G nbsp can be replaced by G 1 displaystyle mathcal G 1 nbsp without changing the topology on F displaystyle F nbsp 2 One may also replace G displaystyle mathcal G nbsp with the collection of all subsets of all finite unions of elements of G displaystyle mathcal G nbsp without changing the resulting G displaystyle mathcal G nbsp topology on F displaystyle F nbsp 4 Call a subset B displaystyle B nbsp of T displaystyle T nbsp F displaystyle F nbsp bounded if f B displaystyle f B nbsp is a bounded subset of Y displaystyle Y nbsp for every f F displaystyle f in F nbsp 5 Theorem 2 5 The G displaystyle mathcal G nbsp topology on F displaystyle F nbsp is compatible with the vector space structure of F displaystyle F nbsp if and only if every G G displaystyle G in mathcal G nbsp is F displaystyle F nbsp bounded that is if and only if for every G G displaystyle G in mathcal G nbsp and every f F displaystyle f in F nbsp f G displaystyle f G nbsp is bounded in Y displaystyle Y nbsp PropertiesProperties of the basic open sets will now be described so assume that G G displaystyle G in mathcal G nbsp and N N displaystyle N in mathcal N nbsp Then U G N displaystyle mathcal U G N nbsp is an absorbing subset of F displaystyle F nbsp if and only if for all f F displaystyle f in F nbsp N displaystyle N nbsp absorbs f G displaystyle f G nbsp 6 If N displaystyle N nbsp is balanced 6 respectively convex then so is U G N displaystyle mathcal U G N nbsp The equality U N F displaystyle mathcal U varnothing N F nbsp always holds If s displaystyle s nbsp is a scalar then s U G N U G s N displaystyle s mathcal U G N mathcal U G sN nbsp so that in particular U G N U G N displaystyle mathcal U G N mathcal U G N nbsp 6 Moreover 4 U G N U G N U G N N displaystyle mathcal U G N mathcal U G N subseteq mathcal U G N N nbsp and similarly 5 U G M U G N U G M N displaystyle mathcal U G M mathcal U G N subseteq mathcal U G M N nbsp For any subsets G H X displaystyle G H subseteq X nbsp and any non empty subsets M N Y displaystyle M N subseteq Y nbsp 5 U G H M N U G M U H N displaystyle mathcal U G cup H M cap N subseteq mathcal U G M cap mathcal U H N nbsp which implies if M N displaystyle M subseteq N nbsp then U G M U G N displaystyle mathcal U G M subseteq mathcal U G N nbsp 6 if G H displaystyle G subseteq H nbsp then U H N U G N displaystyle mathcal U H N subseteq mathcal U G N nbsp For any M N N displaystyle M N in mathcal N nbsp and subsets G H K displaystyle G H K nbsp of T displaystyle T nbsp if G H K displaystyle G cup H subseteq K nbsp then U K M N U G M U H N displaystyle mathcal U K M cap N subseteq mathcal U G M cap mathcal U H N nbsp For any family S displaystyle mathcal S nbsp of subsets of T displaystyle T nbsp and any family M displaystyle mathcal M nbsp of neighborhoods of the origin in Y displaystyle Y nbsp 4 U S S S N S S U S N and U G M M M M M U G M displaystyle 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 nbsp Uniform structure edit See also Uniform space For any G T displaystyle G subseteq T nbsp and U Y Y displaystyle U subseteq Y times Y nbsp be any entourage of Y displaystyle Y nbsp where Y displaystyle Y nbsp is endowed with its canonical uniformity letW G U u v Y T Y T u g v g U for every g G displaystyle 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 nbsp Given G T displaystyle G subseteq T nbsp the family of all sets W G U displaystyle mathcal W G U nbsp as U displaystyle U nbsp ranges over any fundamental system of entourages of Y displaystyle Y nbsp forms a fundamental system of entourages for a uniform structure on Y T displaystyle Y T nbsp called the uniformity of uniform converges on G displaystyle G nbsp or simply the G displaystyle G nbsp convergence uniform structure 7 The G displaystyle mathcal G nbsp convergence uniform structure is the least upper bound of all G displaystyle G nbsp convergence uniform structures as G G displaystyle G in mathcal G nbsp ranges over G displaystyle mathcal G nbsp 7 Nets and uniform convergenceLet f F displaystyle f in F nbsp and let f f i i I displaystyle f bullet left f i right i in I nbsp be a net in F displaystyle F nbsp Then for any subset G displaystyle G nbsp of T displaystyle T nbsp say that f displaystyle f bullet nbsp converges uniformly to f displaystyle f nbsp on G displaystyle G nbsp if for every N N displaystyle N in mathcal N nbsp there exists some i 0 I displaystyle i 0 in I nbsp such that for every i I displaystyle i in I nbsp satisfying i i 0 I displaystyle i geq i 0 I nbsp f i f U G N displaystyle f i f in mathcal U G N nbsp or equivalently f i g f g N displaystyle f i g f g in N nbsp for every g G displaystyle g in G nbsp 5 Theorem 5 If f F displaystyle f in F nbsp and if f f i i I displaystyle f bullet left f i right i in I nbsp is a net in F displaystyle F nbsp then f f displaystyle f bullet to f nbsp in the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp if and only if for every G G displaystyle G in mathcal G nbsp f displaystyle f bullet nbsp converges uniformly to f displaystyle f nbsp on G displaystyle G nbsp Inherited properties edit Local convexityIf Y displaystyle Y nbsp is locally convex then so is the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp and if p i i I displaystyle left p i right i in I nbsp is a family of continuous seminorms generating this topology on Y displaystyle Y nbsp then the G displaystyle mathcal G nbsp topology is induced by the following family of seminorms p G i f sup x G p i f x displaystyle p G i f sup x in G p i f x nbsp as G displaystyle G nbsp varies over G displaystyle mathcal G nbsp and i displaystyle i nbsp varies over I displaystyle I nbsp 8 HausdorffnessIf Y displaystyle Y nbsp is Hausdorff and T G G G displaystyle T bigcup G in mathcal G G nbsp then the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp is Hausdorff 5 Suppose that T displaystyle T nbsp is a topological space If Y displaystyle Y nbsp is Hausdorff and F displaystyle F nbsp is the vector subspace of Y T displaystyle Y T nbsp consisting of all continuous maps that are bounded on every G G displaystyle G in mathcal G nbsp and if G G G displaystyle bigcup G in mathcal G G nbsp is dense in T displaystyle T nbsp then the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp is Hausdorff BoundednessA subset H displaystyle H nbsp of F displaystyle F nbsp is bounded in the G displaystyle mathcal G nbsp topology if and only if for every G G displaystyle G in mathcal G nbsp H G h H h G displaystyle H G bigcup h in H h G nbsp is bounded in Y displaystyle Y nbsp 8 Examples of 𝒢 topologies edit Pointwise convergenceIf we let G displaystyle mathcal G nbsp be the set of all finite subsets of T displaystyle T nbsp then the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp is called the topology of pointwise convergence The topology of pointwise convergence on F displaystyle F nbsp is identical to the subspace topology that F displaystyle F nbsp inherits from Y T displaystyle Y T nbsp when Y T displaystyle Y T nbsp is endowed with the usual product topology If X displaystyle X nbsp is a non trivial completely regular Hausdorff topological space and C X displaystyle C X nbsp is the space of all real or complex valued continuous functions on X displaystyle X nbsp the topology of pointwise convergence on C X displaystyle C X nbsp is metrizable if and only if X displaystyle X nbsp is countable 5 𝒢 topologies on spaces of continuous linear maps editThroughout this section we will assume that X displaystyle X nbsp and Y displaystyle Y nbsp are topological vector spaces G displaystyle mathcal G nbsp will be a non empty collection of subsets of X displaystyle X nbsp directed by inclusion L X Y displaystyle L X Y nbsp will denote the vector space of all continuous linear maps from X displaystyle X nbsp into Y displaystyle Y nbsp If L X Y displaystyle L X Y nbsp is given the G displaystyle mathcal G nbsp topology inherited from Y X displaystyle Y X nbsp then this space with this topology is denoted by L G X Y displaystyle L mathcal G X Y nbsp The continuous dual space of a topological vector space X displaystyle X nbsp over the field F displaystyle mathbb F nbsp which we will assume to be real or complex numbers is the vector space L X F displaystyle L X mathbb F nbsp and is denoted by X displaystyle X prime nbsp The G displaystyle mathcal G nbsp topology on L X Y displaystyle L X Y nbsp is compatible with the vector space structure of L X Y displaystyle L X Y nbsp if and only if for all G G displaystyle G in mathcal G nbsp and all f L X Y displaystyle f in L X Y nbsp the set f G displaystyle f G nbsp is bounded in Y displaystyle Y nbsp which we will assume to be the case for the rest of the article Note in particular that this is the case if G displaystyle mathcal G nbsp consists of von Neumann bounded subsets of X displaystyle X nbsp Assumptions on 𝒢 edit Assumptions that guarantee a vector topology G displaystyle mathcal G nbsp is directed G displaystyle mathcal G nbsp will be a non empty collection of subsets of X displaystyle X nbsp directed by subset inclusion That is for any G H G displaystyle G H in mathcal G nbsp there exists K G displaystyle K in mathcal G nbsp such that G H K displaystyle G cup H subseteq K nbsp The above assumption guarantees that the collection of sets U G N displaystyle mathcal U G N nbsp forms a filter base The next assumption will guarantee that the sets U G N displaystyle mathcal U G N nbsp are balanced Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn t burdensome N N displaystyle N in mathcal N nbsp are balanced N displaystyle mathcal N nbsp is a neighborhoods basis of the origin in Y displaystyle Y nbsp that consists entirely of balanced sets The following assumption is very commonly made because it will guarantee that each set U G N displaystyle mathcal U G N nbsp is absorbing in L X Y displaystyle L X Y nbsp G G displaystyle G in mathcal G nbsp are bounded G displaystyle mathcal G nbsp is assumed to consist entirely of bounded subsets of X displaystyle X nbsp The next theorem gives ways in which G displaystyle mathcal G nbsp can be modified without changing the resulting G displaystyle mathcal G nbsp topology on Y displaystyle Y nbsp Theorem 6 Let G displaystyle mathcal G nbsp be a non empty collection of bounded subsets of X displaystyle X nbsp Then the G displaystyle mathcal G nbsp topology on L X Y displaystyle L X Y nbsp is not altered if G displaystyle mathcal G nbsp is replaced by any of the following collections of also bounded subsets of X displaystyle X nbsp all subsets of all finite unions of sets in G displaystyle mathcal G nbsp all scalar multiples of all sets in G displaystyle mathcal G nbsp all finite Minkowski sums of sets in G displaystyle mathcal G nbsp the balanced hull of every set in G displaystyle mathcal G nbsp the closure of every set in G displaystyle mathcal G nbsp and if X displaystyle X nbsp and Y displaystyle Y nbsp are locally convex then we may add to this list the closed convex balanced hull of every set in G displaystyle mathcal G nbsp Common assumptionsSome authors e g Narici require that G displaystyle mathcal G nbsp satisfy the following condition which implies in particular that G displaystyle mathcal G nbsp is directed by subset inclusion G displaystyle mathcal G nbsp is assumed to be closed with respect to the formation of subsets of finite unions of sets in G displaystyle mathcal G nbsp i e every subset of every finite union of sets in G displaystyle mathcal G nbsp belongs to G displaystyle mathcal G nbsp Some authors e g Treves 9 require that G displaystyle mathcal G nbsp be directed under subset inclusion and that it satisfy the following condition If G G displaystyle G in mathcal G nbsp and s displaystyle s nbsp is a scalar then there exists a H G displaystyle H in mathcal G nbsp such that s G H displaystyle sG subseteq H nbsp If G displaystyle mathcal G nbsp is a bornology on X displaystyle X nbsp which is often the case then these axioms are satisfied If G displaystyle mathcal G nbsp is a saturated family of bounded subsets of X displaystyle X nbsp then these axioms are also satisfied Properties edit HausdorffnessA subset of a TVS X displaystyle X nbsp whose linear span is a dense subset of X displaystyle X nbsp is said to be a total subset of X displaystyle X nbsp If G displaystyle mathcal G nbsp is a family of subsets of a TVS T displaystyle T nbsp then G displaystyle mathcal G nbsp is said to be total in T displaystyle T nbsp if the linear span of G G G displaystyle bigcup G in mathcal G G nbsp is dense in T displaystyle T nbsp 10 If F displaystyle F nbsp is the vector subspace of Y T displaystyle Y T nbsp consisting of all continuous linear maps that are bounded on every G G displaystyle G in mathcal G nbsp then the G displaystyle mathcal G nbsp topology on F displaystyle F nbsp is Hausdorff if Y displaystyle Y nbsp is Hausdorff and G displaystyle mathcal G nbsp is total in T displaystyle T nbsp 6 CompletenessFor the following theorems suppose that X displaystyle X nbsp is a topological vector space and Y displaystyle Y nbsp is a locally convex Hausdorff spaces and G displaystyle mathcal G nbsp is a collection of bounded subsets of X displaystyle X nbsp that covers X displaystyle X nbsp is directed by subset inclusion and satisfies the following condition if G G displaystyle G in mathcal G nbsp and s displaystyle s nbsp is a scalar then there exists a H G displaystyle H in mathcal G nbsp such that s G H displaystyle sG subseteq H nbsp L G X Y displaystyle L mathcal G X Y nbsp is complete if X displaystyle X nbsp is locally convex and Hausdorff Y displaystyle Y nbsp is complete andwhenever u X Y displaystyle u X to Y nbsp is a linear map then u displaystyle u nbsp restricted to every set G G displaystyle G in mathcal G nbsp is continuous implies that u displaystyle u nbsp is continuous If X displaystyle X nbsp is a Mackey space then L G X Y displaystyle L mathcal G X Y nbsp is complete if and only if both X G displaystyle X mathcal G prime nbsp and Y displaystyle Y nbsp are complete If X displaystyle X nbsp is barrelled then L G X Y displaystyle L mathcal G X Y nbsp is Hausdorff and quasi complete Let X displaystyle X nbsp and Y displaystyle Y nbsp be TVSs with Y displaystyle Y nbsp quasi complete and assume that 1 X displaystyle X nbsp is barreled or else 2 X displaystyle X nbsp is a Baire space and X displaystyle X nbsp and Y displaystyle Y nbsp are locally convex If G displaystyle mathcal G nbsp covers X displaystyle X nbsp then every closed equicontinuous subset of L X Y displaystyle L X Y nbsp is complete in L G X Y displaystyle L mathcal G X Y nbsp and L G X Y displaystyle L mathcal G X Y nbsp is quasi complete 11 Let X displaystyle X nbsp be a bornological space Y displaystyle Y nbsp a locally convex space and G displaystyle mathcal G nbsp a family of bounded subsets of X displaystyle X nbsp such that the range of every null sequence in X displaystyle X nbsp is contained in some G G displaystyle G in mathcal G nbsp If Y displaystyle Y nbsp is quasi complete respectively complete then so is L G X Y displaystyle L mathcal G X Y nbsp 12 BoundednessLet X displaystyle X nbsp and Y displaystyle Y nbsp be topological vector spaces and H displaystyle H nbsp be a subset of L X Y displaystyle L X Y nbsp Then the following are equivalent 8 H displaystyle H nbsp is bounded in L G X Y displaystyle L mathcal G X Y nbsp For every G G displaystyle G in mathcal G nbsp H G h H h G displaystyle H G bigcup h in H h G nbsp is bounded in Y displaystyle Y nbsp 8 For every neighborhood V displaystyle V nbsp of the origin in Y displaystyle Y nbsp the set h H h 1 V displaystyle bigcap h in H h 1 V nbsp absorbs every G G displaystyle G in mathcal G nbsp If G displaystyle mathcal G nbsp is a collection of bounded subsets of X displaystyle X nbsp whose union is total in X displaystyle X nbsp then every equicontinuous subset of L X Y displaystyle L X Y nbsp is bounded in the G displaystyle mathcal G nbsp topology 11 Furthermore if X displaystyle X nbsp and Y displaystyle Y nbsp are locally convex Hausdorff spaces then if H displaystyle H nbsp is bounded in L s X Y displaystyle L sigma X Y nbsp 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 X displaystyle X nbsp 13 if X displaystyle X nbsp is quasi complete meaning that closed and bounded subsets are complete then the bounded subsets of L X Y displaystyle L X Y nbsp are identical for all G displaystyle mathcal G nbsp topologies where G displaystyle mathcal G nbsp is any family of bounded subsets of X displaystyle X nbsp covering X displaystyle X nbsp 13 Examples edit G X displaystyle mathcal G subseteq wp X nbsp topology of uniform convergence on Notation Name topology of Alternative namefinite subsets of X displaystyle X nbsp L s X Y displaystyle L sigma X Y nbsp pointwise simple convergence topology of simple convergenceprecompact subsets of X displaystyle X nbsp precompact convergencecompact convex subsets of X displaystyle X nbsp L g X Y displaystyle L gamma X Y nbsp compact convex convergencecompact subsets of X displaystyle X nbsp L c X Y displaystyle L c X Y nbsp compact convergencebounded subsets of X displaystyle X nbsp L b X Y displaystyle L b X Y nbsp bounded convergence strong topologyThe topology of pointwise convergence edit By letting G displaystyle mathcal G nbsp be the set of all finite subsets of X displaystyle X nbsp L X Y displaystyle L X Y nbsp will have the weak topology on L X Y displaystyle L X Y nbsp or the topology of pointwise convergence or the topology of simple convergence and L X Y displaystyle L X Y nbsp with this topology is denoted by L s X Y displaystyle L sigma X Y nbsp Unfortunately this topology is also sometimes called the strong operator topology which may lead to ambiguity 6 for this reason this article will avoid referring to this topology by this name A subset of L X Y displaystyle L X Y nbsp is called simply bounded or weakly bounded if it is bounded in L s X Y displaystyle L sigma X Y nbsp The weak topology on L X Y displaystyle L X Y nbsp has the following properties If X displaystyle X nbsp is separable that is it has a countable dense subset and if Y displaystyle Y nbsp is a metrizable topological vector space then every equicontinuous subset H displaystyle H nbsp of L s X Y displaystyle L sigma X Y nbsp is metrizable if in addition Y displaystyle Y nbsp is separable then so is H displaystyle H nbsp 14 So in particular on every equicontinuous subset of L X Y displaystyle L X Y nbsp the topology of pointwise convergence is metrizable Let Y X displaystyle Y X nbsp denote the space of all functions from X displaystyle X nbsp into Y displaystyle Y nbsp If L X Y displaystyle L X Y nbsp is given the topology of pointwise convergence then space of all linear maps continuous or not X displaystyle X nbsp into Y displaystyle Y nbsp is closed in Y X displaystyle Y X nbsp In addition L X Y displaystyle L X Y nbsp is dense in the space of all linear maps continuous or not X displaystyle X nbsp into Y displaystyle Y nbsp Suppose X displaystyle X nbsp and Y displaystyle Y nbsp are locally convex Any simply bounded subset of L X Y displaystyle L X Y nbsp is bounded when L X Y displaystyle L X Y nbsp has the topology of uniform convergence on convex balanced bounded complete subsets of X displaystyle X nbsp If in addition X displaystyle X nbsp is quasi complete then the families of bounded subsets of L X Y displaystyle L X Y nbsp are identical for all G displaystyle mathcal G nbsp topologies on L X Y displaystyle L X Y nbsp such that G displaystyle mathcal G nbsp is a family of bounded sets covering X displaystyle X nbsp 13 Equicontinuous subsets The weak closure of an equicontinuous subset of L X Y displaystyle L X Y nbsp is equicontinuous If Y displaystyle Y nbsp is locally convex then the convex balanced hull of an equicontinuous subset of L X Y displaystyle L X Y nbsp is equicontinuous Let X displaystyle X nbsp and Y displaystyle Y nbsp be TVSs and assume that 1 X displaystyle X nbsp is barreled or else 2 X displaystyle X nbsp is a Baire space and X displaystyle X nbsp and Y displaystyle Y nbsp are locally convex Then every simply bounded subset of L X Y displaystyle L X Y nbsp is equicontinuous 11 On an equicontinuous subset H displaystyle H nbsp of L X Y displaystyle L X Y nbsp the following topologies are identical 1 topology of pointwise convergence on a total subset of X displaystyle X nbsp 2 the topology of pointwise convergence 3 the topology of precompact convergence 11 Compact convergence edit By letting G displaystyle mathcal G nbsp be the set of all compact subsets of X displaystyle X nbsp L X Y displaystyle L X Y nbsp will have the topology of compact convergence or the topology of uniform convergence on compact sets and L X Y displaystyle L X Y nbsp with this topology is denoted by L c X Y displaystyle L c X Y nbsp The topology of compact convergence on L X Y displaystyle L X Y nbsp has the following properties If X displaystyle X nbsp is a Frechet space or a LF space and if Y displaystyle Y nbsp is a complete locally convex Hausdorff space then L c X Y displaystyle L c X Y nbsp is complete On equicontinuous subsets of L X Y displaystyle L X Y nbsp the following topologies coincide The topology of pointwise convergence on a dense subset of X displaystyle X nbsp The topology of pointwise convergence on X displaystyle X nbsp The topology of compact convergence The topology of precompact convergence If X displaystyle X nbsp is a Montel space and Y displaystyle Y nbsp is a topological vector space then L c X Y displaystyle L c X Y nbsp and L b X Y displaystyle L b X Y nbsp have identical topologies Topology of bounded convergence edit By letting G displaystyle mathcal G nbsp be the set of all bounded subsets of X displaystyle X nbsp L X Y displaystyle L X Y nbsp will have the topology of bounded convergence on X displaystyle X nbsp or the topology of uniform convergence on bounded sets and L X Y displaystyle L X Y nbsp with this topology is denoted by L b X Y displaystyle L b X Y nbsp 6 The topology of bounded convergence on L X Y displaystyle L X Y nbsp has the following properties If X displaystyle X nbsp is a bornological space and if Y displaystyle Y nbsp is a complete locally convex Hausdorff space then L b X Y displaystyle L b X Y nbsp is complete If X displaystyle X nbsp and Y displaystyle Y nbsp are both normed spaces then the topology on L X Y displaystyle L X Y nbsp induced by the usual operator norm is identical to the topology on L b X Y displaystyle L b X Y nbsp 6 In particular if X displaystyle X nbsp is a normed space then the usual norm topology on the continuous dual space X displaystyle X prime nbsp is identical to the topology of bounded convergence on X displaystyle X prime nbsp Every equicontinuous subset of L X Y displaystyle L X Y nbsp is bounded in L b X Y displaystyle L b X Y nbsp Polar topologies editMain article Polar topology Throughout we assume that X displaystyle X nbsp is a TVS 𝒢 topologies versus polar topologies edit If X displaystyle X nbsp is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets e g if X displaystyle X nbsp is a Hausdorff locally convex space then a G displaystyle mathcal G nbsp topology on X displaystyle X prime nbsp as defined in this article is a polar topology and conversely every polar topology if a G displaystyle mathcal G nbsp topology Consequently in this case the results mentioned in this article can be applied to polar topologies However if X displaystyle X nbsp is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets then the notion of bounded in X displaystyle X nbsp is stronger than the notion of s X X displaystyle sigma left X X prime right nbsp bounded in X displaystyle X nbsp i e bounded in X displaystyle X nbsp implies s X X displaystyle sigma left X X prime right nbsp bounded in X displaystyle X nbsp so that a G displaystyle mathcal G nbsp topology on X displaystyle X prime nbsp as defined in this article is not necessarily a polar topology One important difference is that polar topologies are always locally convex while G displaystyle mathcal G nbsp topologies need not be Polar topologies have stronger results than the more general topologies of uniform convergence described, wikipedia, wiki, book, books, library,