Topologies on spaces of linear maps

In mathematics, a linear map is a mapping between two modules (including vector spaces) 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.
Throughout we assume the following:
Basic neighborhoods at the origin.
Henceforth assume that and .
-topology.
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 (i.e. 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 details).
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 .
Inherited properties.
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 .
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.
A subset of is bounded in the -topology if and only if for every , is bounded in .
Examples of 𝒢-topologies.
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.
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.
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 .
The next theorem gives ways in which can be modified without changing the resulting -topology on .
Some authors (e.g. Narici) require that satisfy the following condition, which implies, in particular, that is directed by subset inclusion: 
Some authors (e.g. Trèves) 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.
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 .
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 and be topological vector spaces and be a subset of . 
Then the following are equivalent:
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: 
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 (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 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.
-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 (either the real or complex numbers). 
In an analogous way to how we placed a topology on we can place a topology on and .
Let (resp. ) be a family of subsets of (resp. ) 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 (that is, in or in ) 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:
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 formula_1 will be a topological vector space topology. 
This topology is called the ε-topology and formula_2 with this topology it is denoted by formula_3 or simply by formula_4.
Part of the importance of this vector space and this topology is that it contains many subspace, such as formula_5, which we denote by formula_6. 
When this subspace is given the subspace topology of formula_4 it is denoted by formula_8.
In the instance where is the field of these vector spaces, formula_9 is a tensor product of and . 
In fact, if and are locally convex Hausdorff spaces then formula_9 is vector space-isomorphic to formula_11, which is in turn is equal to formula_12.
These spaces have the following properties: