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).
Topologies of uniform convergence on arbitrary spaces of maps
Throughout, the following is assumed: is any non-empty set and is a non-empty collection of subsets of directed by subset inclusion (i.e. for any there exists some such that ). is a topological vector space (not necessarily Hausdorff or locally convex). is a basis of neighborhoods of 0 in is a vector subspace of Because is just a set that is not yet assumed to be endowed with any vector space structure, should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined. which denotes the set of all -valued functions with domain
𝒢-topology
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 basisNote that each set is a neighborhood of the origin for this topology, but it is not necessarily an open neighborhood of the origin. at the origin for a unique translation-invariant topology on where this topology is 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. 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 detailsIn practice, 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, is the collection of compact subsets of (and is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of ).
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 One may also replace with the collection of all subsets of all finite unions of elements of without changing the resulting -topology on
Call a subset of -bounded if is a bounded subset of for every
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 . If is balanced (respectively, convex) then so is
The equality
always holds. If is a scalar then so that in particular, Moreover,
and similarly
For any subsets and any non-empty subsets
which implies: if then if then For any and subsets of if then
For any family of subsets of and any family of neighborhoods of the origin in
Uniform structure
For any and be any entourage of (where is endowed with its canonical uniformity), let
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 or simply . The is the least upper bound of all -convergence uniform structures as ranges over
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 ).
Inherited properties
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:
as varies over and varies over .
Hausdorffness
If is Hausdorff and then the -topology on is Hausdorff.
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
Examples of 𝒢-topologies
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.
𝒢-topologies on spaces of continuous linear maps
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 𝒢
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
Common assumptions
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 ) 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.
Properties
Hausdorffness
A subset of a TVS whose linear span is a dense subset of is said to be a total subset of If is a family of subsets of a TVS then is said to be total in if the linear span of is dense in
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
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
is complete if If is a Mackey space then is complete if and only if both and are complete. If is barrelled then is Hausdorff and quasi-complete. Let and be TVSs with quasi-complete and assume that (1) is barreled, or else (2) is a Baire space and and are locally convex. If covers then every closed equicontinuous subset of is complete in and is quasi-complete. 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 .
Boundedness
Let and be topological vector spaces and be a subset of Then the following are equivalent: is bounded in ; For every is bounded in ; 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. 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 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
Examples
("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 convergence
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; 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 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
Equicontinuous subsets
The weak-closure of an equicontinuous subset of is equicontinuous. 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. 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.
Compact convergence
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: