Topologies on spaces of linear maps

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.

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.
is a topological vector space.
is a basis of neighborhoods of 0 in
is a vector subspace of which denotes the set of all -valued functions with domain

?-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 basis
at the origin for a unique translation-invariant topology on where this topology is necessarily a vector topology.
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 .
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 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, 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 .

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 ?-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 valued continuous functions on the topology of pointwise convergence on is metrizable if and only if is countable.

?-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 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 ?

Assumptions that guarantee a vector topology

: will be a non-empty collection of subsets of directed by 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.

: 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

: 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 require that satisfy the following condition, which implies, in particular, that is directed by subset inclusion:
Some authors require that be directed under subset inclusion and that it satisfy the following condition:
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 is barreled, or else 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 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 then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of
if is quasi-complete, then the bounded subsets of are identical for all -topologies where is any family of bounded subsets of covering

Examples

	Notation	Name	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 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 into is closed in.
- In addition, is dense in the space of all linear maps 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 is barreled, or else 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: topology of pointwise convergence on a total subset of ; the topology of pointwise convergence; 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:

If is a Fréchet space or a LF-space and if is a complete locally convex Hausdorff space then is complete.
On equicontinuous subsets of the following topologies coincide:
- 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 convergence

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.
The topology of bounded convergence on has the following properties:

If is a bornological space and if is a complete locally convex Hausdorff space then is complete.
If and are both normed spaces then the topology on induced by the usual operator norm is identical to the topology on.
- In particular, if is a normed space then the usual norm topology on the continuous dual space is identical to the topology of bounded convergence on.
Every equicontinuous subset of is bounded in.

Polar topologies

Throughout, we assume that is a TVS.

?-topologies versus polar topologies

If is a TVS whose bounded subsets are exactly the same as its bounded subsets, then a -topology on 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 exactly the same as its bounded subsets, then the notion of "bounded in " is stronger than the notion of "-bounded in " so that a -topology on is 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 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 is a TVS whose bounded subsets are the same as its weakly bounded subsets.
Notation: If denotes a polar topology on then endowed with this topology will be denoted by or simply .

>	Notation	Name	Alternative name
finite subsets of		pointwise/simple convergence	weak/weak* topology
-compact disks			Mackey topology
-compact convex subsets		compact convex convergence
-compact subsets		compact convergence
-bounded subsets		bounded convergence	strong topology

?-ℋ topologies on spaces of bilinear maps

We will let denote the space of separately continuous bilinear maps and denote the space of continuous bilinear maps, where and are topological vector space over the same field.
In an analogous way to how we placed a topology on we can place a topology on and.
Let be a family of subsets of containing at least one non-empty set.
Let denote the collection of all sets where
We can place on the -topology, and consequently on any of its subsets, in particular on and on.
This topology is known as the -topology or as the topology of uniform convergence on the products of .
However, as before, this topology is not necessarily compatible with the vector space structure of or of without the additional requirement that for all bilinear maps, in this space and for all and the set is bounded in
If both and consist of bounded sets then this requirement is automatically satisfied if we are topologizing but this may not be the case if we are trying to topologize.
The -topology on will be compatible with the vector space structure of if both and consists of bounded sets and any of the following conditions hold:

and are barrelled spaces and is locally convex.
is a F-space, is metrizable, and is Hausdorff, in which case
and are the strong duals of reflexive Fréchet spaces.
is normed and and the strong duals of reflexive Fréchet spaces.

The ε-topology

Suppose that and are locally convex spaces and let and be the collections of equicontinuous subsets of and, respectively.
Then the -topology on will be a topological vector space topology.
This topology is called the ε-topology and with this topology it is denoted by or simply by
Part of the importance of this vector space and this topology is that it contains many subspace, such as which we denote by
When this subspace is given the subspace topology of it is denoted by
In the instance where is the field of these vector spaces, is a tensor product of and
In fact, if and are locally convex Hausdorff spaces then is vector space-isomorphic to which is in turn is equal to
These spaces have the following properties:

If and are locally convex Hausdorff spaces then is complete if and only if both and are complete.
If and are both normed then so is