Topologies on spaces of linear maps


In mathematics, a linear map is a mapping between two modules that preserves the operations of addition and scalar multiplication.
By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornologies, then one can study the subspace of linear maps that preserve this structure.

Topologies of uniform convergence on arbitrary spaces of maps

Let be any set and be a non-empty collection of subsets of directed by subset inclusion.
Let be a topological vector space and let be a basis of neighborhoods of 0 in.
Let denote the set of all -valued functions with domain and let be a vector subspace of .

Basic neighborhoods at the origin

For all and, let
;Properties
;Algebraic relations

-topology

Assume is directed by subset inclusion.
Then the set forms a neighborhood basis
at the origin for a unique translation-invariant topology on, where this topology is not 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.
The -topology on is compatible with the vector space structure of if and only if for every and every, is bounded in.

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
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.

-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.
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 bounded subsets of.

Assumptions on

;Assumptions that guarantee a vector topology
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 burdonsome.
The following assumption is very commonly made because it will guarantee that each set is absorbing in.
;Other possible assumptions
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
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.
;Boundedness
Let and be topological vector spaces and be a subset of.
Then the following are equivalent:

  1. is bounded in ;
  2. For every, is bounded in ;
  3. For every neighborhood of 0 in the set absorbs every.
Furthermore,

Examples

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.
The weak-topology on has the following properties:
;Equicontinuous subsets

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:

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:

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 weakly 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 not exactly the same as its weakly bounded subsets, then the notion of "bounded in " is stronger than the notion of "-bounded in " so that a -topology on 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 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.

NotationName Alternative name
finite subsets of
pointwise/simple convergenceweak/weak* topology
-compact disksMackey topology
-compact convex subsetscompact convex convergence
-compact subsets
compact convergence
-bounded subsets
bounded convergencestrong topology

-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:
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: