Throughout, the following is assumed:

1. ${\displaystyle T}$  is any non-empty set and ${\displaystyle {\mathcal {G}}}$  is a non-empty collection of subsets of ${\displaystyle T}$  directed by subset inclusion (i.e. for any ${\displaystyle G,H\in {\mathcal {G}}}$  there exists some ${\displaystyle K\in {\mathcal {G}}}$  such that ${\displaystyle G\cup H\subseteq K}$ ).
2. ${\displaystyle Y}$  is a topological vector space (not necessarily Hausdorff or locally convex).
3. ${\displaystyle {\mathcal {N}}}$  is a basis of neighborhoods of 0 in ${\displaystyle Y.}$ 
4. ${\displaystyle F}$  is a vector subspace of ${\displaystyle Y^{T}=\prod _{t\in T}Y,}$ ^{[note 1]} which denotes the set of all ${\displaystyle Y}$ -valued functions ${\displaystyle f:T\to Y}$  with domain ${\displaystyle T.}$ 

𝒢-topology edit

The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets ${\displaystyle G\subseteq T}$  and ${\displaystyle N\subseteq Y,}$  let

The family

A subset ${\displaystyle {\mathcal {G}}_{1}}$  of ${\displaystyle {\mathcal {G}}}$  is said to be fundamental with respect to ${\displaystyle {\mathcal {G}}}$  if each ${\displaystyle G\in {\mathcal {G}}}$  is a subset of some element in ${\displaystyle {\mathcal {G}}_{1}.}$  In this case, the collection ${\displaystyle {\mathcal {G}}}$  can be replaced by ${\displaystyle {\mathcal {G}}_{1}}$  without changing the topology on ${\displaystyle F.}$ ^[2] One may also replace ${\displaystyle {\mathcal {G}}}$  with the collection of all subsets of all finite unions of elements of ${\displaystyle {\mathcal {G}}}$  without changing the resulting ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F.}$ ^[4]

Call a subset ${\displaystyle B}$  of ${\displaystyle T}$  ${\displaystyle F}$ -bounded if ${\displaystyle f(B)}$  is a bounded subset of ${\displaystyle Y}$  for every ${\displaystyle f\in F.}$ ^[5]

Properties

Properties of the basic open sets will now be described, so assume that ${\displaystyle G\in {\mathcal {G}}}$  and ${\displaystyle N\in {\mathcal {N}}.}$  Then ${\displaystyle {\mathcal {U}}(G,N)}$  is an absorbing subset of ${\displaystyle F}$  if and only if for all ${\displaystyle f\in F,}$  ${\displaystyle N}$  absorbs ${\displaystyle f(G)}$ .^[6] If ${\displaystyle N}$  is balanced^[6] (respectively, convex) then so is ${\displaystyle {\mathcal {U}}(G,N).}$ 

The equality ${\displaystyle {\mathcal {U}}(\varnothing ,N)=F}$  always holds. If ${\displaystyle s}$  is a scalar then ${\displaystyle s{\mathcal {U}}(G,N)={\mathcal {U}}(G,sN),}$  so that in particular, ${\displaystyle -{\mathcal {U}}(G,N)={\mathcal {U}}(G,-N).}$ ^[6] Moreover,^[4]

For any subsets ${\displaystyle G,H\subseteq X}$  and any non-empty subsets ${\displaystyle M,N\subseteq Y,}$ ^[5]

• if ${\displaystyle M\subseteq N}$  then ${\displaystyle {\mathcal {U}}(G,M)\subseteq {\mathcal {U}}(G,N).}$ ^[6]
• if ${\displaystyle G\subseteq H}$  then ${\displaystyle {\mathcal {U}}(H,N)\subseteq {\mathcal {U}}(G,N).}$ 
• For any ${\displaystyle M,N\in {\mathcal {N}}}$  and subsets ${\displaystyle G,H,K}$  of ${\displaystyle T,}$  if ${\displaystyle G\cup H\subseteq K}$  then ${\displaystyle {\mathcal {U}}(K,M\cap N)\subseteq {\mathcal {U}}(G,M)\cap {\mathcal {U}}(H,N).}$ 

For any family ${\displaystyle {\mathcal {S}}}$  of subsets of ${\displaystyle T}$  and any family ${\displaystyle {\mathcal {M}}}$  of neighborhoods of the origin in ${\displaystyle Y,}$ ^[4]

Uniform structure edit

For any ${\displaystyle G\subseteq T}$  and ${\displaystyle U\subseteq Y\times Y}$  be any entourage of ${\displaystyle Y}$  (where ${\displaystyle Y}$  is endowed with its canonical uniformity), let

Nets and uniform convergence

Let ${\displaystyle f\in F}$  and let ${\displaystyle f_{\bullet }=\left(f_{i}\right)_{i\in I}}$  be a net in ${\displaystyle F.}$  Then for any subset ${\displaystyle G}$  of ${\displaystyle T,}$  say that ${\displaystyle f_{\bullet }}$  converges uniformly to ${\displaystyle f}$  on ${\displaystyle G}$  if for every ${\displaystyle N\in {\mathcal {N}}}$  there exists some ${\displaystyle i_{0}\in I}$  such that for every ${\displaystyle i\in I}$  satisfying ${\displaystyle i\geq i_{0},I}$  ${\displaystyle f_{i}-f\in {\mathcal {U}}(G,N)}$  (or equivalently, ${\displaystyle f_{i}(g)-f(g)\in N}$  for every ${\displaystyle g\in G}$ ).^[5]

Inherited properties edit

Local convexity

If ${\displaystyle Y}$  is locally convex then so is the ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F}$  and if ${\displaystyle \left(p_{i}\right)_{i\in I}}$  is a family of continuous seminorms generating this topology on ${\displaystyle Y}$  then the ${\displaystyle {\mathcal {G}}}$ -topology is induced by the following family of seminorms:

Hausdorffness

If ${\displaystyle Y}$  is Hausdorff and ${\displaystyle T=\bigcup _{G\in {\mathcal {G}}}G}$  then the ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F}$  is Hausdorff.^[5]

Suppose that ${\displaystyle T}$  is a topological space. If ${\displaystyle Y}$  is Hausdorff and ${\displaystyle F}$  is the vector subspace of ${\displaystyle Y^{T}}$  consisting of all continuous maps that are bounded on every ${\displaystyle G\in {\mathcal {G}}}$  and if ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$  is dense in ${\displaystyle T}$  then the ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F}$  is Hausdorff.

Boundedness

A subset ${\displaystyle H}$  of ${\displaystyle F}$  is bounded in the ${\displaystyle {\mathcal {G}}}$ -topology if and only if for every ${\displaystyle G\in {\mathcal {G}},}$  ${\displaystyle H(G)=\bigcup _{h\in H}h(G)}$  is bounded in ${\displaystyle Y.}$ ^[8]

Examples of 𝒢-topologies edit

Pointwise convergence

If we let ${\displaystyle {\mathcal {G}}}$  be the set of all finite subsets of ${\displaystyle T}$  then the ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F}$  is called the topology of pointwise convergence. The topology of pointwise convergence on ${\displaystyle F}$  is identical to the subspace topology that ${\displaystyle F}$  inherits from ${\displaystyle Y^{T}}$  when ${\displaystyle Y^{T}}$  is endowed with the usual product topology.

If ${\displaystyle X}$  is a non-trivial completely regular Hausdorff topological space and ${\displaystyle C(X)}$  is the space of all real (or complex) valued continuous functions on ${\displaystyle X,}$  the topology of pointwise convergence on ${\displaystyle C(X)}$  is metrizable if and only if ${\displaystyle X}$  is countable.^[5]

Throughout this section we will assume that ${\displaystyle X}$  and ${\displaystyle Y}$  are topological vector spaces. ${\displaystyle {\mathcal {G}}}$  will be a non-empty collection of subsets of ${\displaystyle X}$  directed by inclusion. ${\displaystyle L(X;Y)}$  will denote the vector space of all continuous linear maps from ${\displaystyle X}$  into ${\displaystyle Y.}$  If ${\displaystyle L(X;Y)}$  is given the ${\displaystyle {\mathcal {G}}}$ -topology inherited from ${\displaystyle Y^{X}}$  then this space with this topology is denoted by ${\displaystyle L_{\mathcal {G}}(X;Y)}$ . The continuous dual space of a topological vector space ${\displaystyle X}$  over the field ${\displaystyle \mathbb {F} }$  (which we will assume to be real or complex numbers) is the vector space ${\displaystyle L(X;\mathbb {F} )}$  and is denoted by ${\displaystyle X^{\prime }}$ .

The ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle L(X;Y)}$  is compatible with the vector space structure of ${\displaystyle L(X;Y)}$  if and only if for all ${\displaystyle G\in {\mathcal {G}}}$  and all ${\displaystyle f\in L(X;Y)}$  the set ${\displaystyle f(G)}$  is bounded in ${\displaystyle Y,}$  which we will assume to be the case for the rest of the article. Note in particular that this is the case if ${\displaystyle {\mathcal {G}}}$  consists of (von-Neumann) bounded subsets of ${\displaystyle X.}$ 

Assumptions on 𝒢 edit

Assumptions that guarantee a vector topology

• (${\displaystyle {\mathcal {G}}}$  is directed): ${\displaystyle {\mathcal {G}}}$  will be a non-empty collection of subsets of ${\displaystyle X}$  directed by (subset) inclusion. That is, for any ${\displaystyle G,H\in {\mathcal {G}},}$  there exists ${\displaystyle K\in {\mathcal {G}}}$  such that ${\displaystyle G\cup H\subseteq K}$ .

The above assumption guarantees that the collection of sets ${\displaystyle {\mathcal {U}}(G,N)}$  forms a filter base. The next assumption will guarantee that the sets ${\displaystyle {\mathcal {U}}(G,N)}$  are balanced. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.

• (${\displaystyle N\in {\mathcal {N}}}$  are balanced): ${\displaystyle {\mathcal {N}}}$  is a neighborhoods basis of the origin in ${\displaystyle Y}$  that consists entirely of balanced sets.

The following assumption is very commonly made because it will guarantee that each set ${\displaystyle {\mathcal {U}}(G,N)}$  is absorbing in ${\displaystyle L(X;Y).}$ 

• (${\displaystyle G\in {\mathcal {G}}}$  are bounded): ${\displaystyle {\mathcal {G}}}$  is assumed to consist entirely of bounded subsets of ${\displaystyle X.}$ 

The next theorem gives ways in which ${\displaystyle {\mathcal {G}}}$  can be modified without changing the resulting ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle Y.}$ 

Common assumptions

Some authors (e.g. Narici) require that ${\displaystyle {\mathcal {G}}}$  satisfy the following condition, which implies, in particular, that ${\displaystyle {\mathcal {G}}}$  is directed by subset inclusion:

${\displaystyle {\mathcal {G}}}$  is assumed to be closed with respect to the formation of subsets of finite unions of sets in ${\displaystyle {\mathcal {G}}}$  (i.e. every subset of every finite union of sets in ${\displaystyle {\mathcal {G}}}$  belongs to ${\displaystyle {\mathcal {G}}}$ ).

Some authors (e.g. Trèves ^[9]) require that ${\displaystyle {\mathcal {G}}}$  be directed under subset inclusion and that it satisfy the following condition:

If ${\displaystyle G\in {\mathcal {G}}}$  and ${\displaystyle s}$  is a scalar then there exists a ${\displaystyle H\in {\mathcal {G}}}$  such that ${\displaystyle sG\subseteq H.}$ 

If ${\displaystyle {\mathcal {G}}}$  is a bornology on ${\displaystyle X,}$  which is often the case, then these axioms are satisfied. If ${\displaystyle {\mathcal {G}}}$  is a saturated family of bounded subsets of ${\displaystyle X}$  then these axioms are also satisfied.

Properties edit

Hausdorffness

A subset of a TVS ${\displaystyle X}$  whose linear span is a dense subset of ${\displaystyle X}$  is said to be a total subset of ${\displaystyle X.}$  If ${\displaystyle {\mathcal {G}}}$  is a family of subsets of a TVS ${\displaystyle T}$  then ${\displaystyle {\mathcal {G}}}$  is said to be total in ${\displaystyle T}$  if the linear span of ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$  is dense in ${\displaystyle T.}$ ^[10]

If ${\displaystyle F}$  is the vector subspace of ${\displaystyle Y^{T}}$  consisting of all continuous linear maps that are bounded on every ${\displaystyle G\in {\mathcal {G}},}$  then the ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle F}$  is Hausdorff if ${\displaystyle Y}$  is Hausdorff and ${\displaystyle {\mathcal {G}}}$  is total in ${\displaystyle T.}$ ^[6]

Completeness

For the following theorems, suppose that ${\displaystyle X}$  is a topological vector space and ${\displaystyle Y}$  is a locally convex Hausdorff spaces and ${\displaystyle {\mathcal {G}}}$  is a collection of bounded subsets of ${\displaystyle X}$  that covers ${\displaystyle X,}$  is directed by subset inclusion, and satisfies the following condition: if ${\displaystyle G\in {\mathcal {G}}}$  and ${\displaystyle s}$  is a scalar then there exists a ${\displaystyle H\in {\mathcal {G}}}$  such that ${\displaystyle sG\subseteq H.}$ 

• ${\displaystyle L_{\mathcal {G}}(X;Y)}$  is complete if
  1. ${\displaystyle X}$  is locally convex and Hausdorff,
  2. ${\displaystyle Y}$  is complete, and
  3. whenever ${\displaystyle u:X\to Y}$  is a linear map then ${\displaystyle u}$  restricted to every set ${\displaystyle G\in {\mathcal {G}}}$  is continuous implies that ${\displaystyle u}$  is continuous,
• If ${\displaystyle X}$  is a Mackey space then ${\displaystyle L_{\mathcal {G}}(X;Y)}$ is complete if and only if both ${\displaystyle X_{\mathcal {G}}^{\prime }}$  and ${\displaystyle Y}$  are complete.
• If ${\displaystyle X}$  is barrelled then ${\displaystyle L_{\mathcal {G}}(X;Y)}$  is Hausdorff and quasi-complete.
• Let ${\displaystyle X}$  and ${\displaystyle Y}$  be TVSs with ${\displaystyle Y}$  quasi-complete and assume that (1) ${\displaystyle X}$  is barreled, or else (2) ${\displaystyle X}$  is a Baire space and ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex. If ${\displaystyle {\mathcal {G}}}$  covers ${\displaystyle X}$  then every closed equicontinuous subset of ${\displaystyle L(X;Y)}$  is complete in ${\displaystyle L_{\mathcal {G}}(X;Y)}$  and ${\displaystyle L_{\mathcal {G}}(X;Y)}$  is quasi-complete.^[11]
• Let ${\displaystyle X}$  be a bornological space, ${\displaystyle Y}$  a locally convex space, and ${\displaystyle {\mathcal {G}}}$  a family of bounded subsets of ${\displaystyle X}$  such that the range of every null sequence in ${\displaystyle X}$  is contained in some ${\displaystyle G\in {\mathcal {G}}.}$  If ${\displaystyle Y}$  is quasi-complete (respectively, complete) then so is ${\displaystyle L_{\mathcal {G}}(X;Y)}$ .^[12]

Boundedness

Let ${\displaystyle X}$  and ${\displaystyle Y}$  be topological vector spaces and ${\displaystyle H}$  be a subset of ${\displaystyle L(X;Y).}$  Then the following are equivalent:^[8]

1. ${\displaystyle H}$  is bounded in ${\displaystyle L_{\mathcal {G}}(X;Y)}$ ;
2. For every ${\displaystyle G\in {\mathcal {G}},}$  ${\displaystyle H(G):=\bigcup _{h\in H}h(G)}$  is bounded in ${\displaystyle Y}$ ;^[8]
3. For every neighborhood ${\displaystyle V}$  of the origin in ${\displaystyle Y}$  the set ${\displaystyle \bigcap _{h\in H}h^{-1}(V)}$  absorbs every ${\displaystyle G\in {\mathcal {G}}.}$ 

If ${\displaystyle {\mathcal {G}}}$  is a collection of bounded subsets of ${\displaystyle X}$  whose union is total in ${\displaystyle X}$  then every equicontinuous subset of ${\displaystyle L(X;Y)}$  is bounded in the ${\displaystyle {\mathcal {G}}}$ -topology.^[11] Furthermore, if ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex Hausdorff spaces then

• if ${\displaystyle H}$  is bounded in ${\displaystyle L_{\sigma }(X;Y)}$  (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 ${\displaystyle X.}$ ^[13]
• if ${\displaystyle X}$  is quasi-complete (meaning that closed and bounded subsets are complete), then the bounded subsets of ${\displaystyle L(X;Y)}$  are identical for all ${\displaystyle {\mathcal {G}}}$ -topologies where ${\displaystyle {\mathcal {G}}}$  is any family of bounded subsets of ${\displaystyle X}$  covering ${\displaystyle X.}$ ^[13]

Examples edit

The topology of pointwise convergence edit

By letting ${\displaystyle {\mathcal {G}}}$  be the set of all finite subsets of ${\displaystyle X,}$  ${\displaystyle L(X;Y)}$  will have the weak topology on ${\displaystyle L(X;Y)}$  or the topology of pointwise convergence or the topology of simple convergence and ${\displaystyle L(X;Y)}$  with this topology is denoted by ${\displaystyle L_{\sigma }(X;Y)}$ . 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 ${\displaystyle L(X;Y)}$  is called simply bounded or weakly bounded if it is bounded in ${\displaystyle L_{\sigma }(X;Y)}$ .

The weak-topology on ${\displaystyle L(X;Y)}$  has the following properties:

• If ${\displaystyle X}$  is separable (that is, it has a countable dense subset) and if ${\displaystyle Y}$  is a metrizable topological vector space then every equicontinuous subset ${\displaystyle H}$  of ${\displaystyle L_{\sigma }(X;Y)}$  is metrizable; if in addition ${\displaystyle Y}$  is separable then so is ${\displaystyle H.}$ ^[14]
  • So in particular, on every equicontinuous subset of ${\displaystyle L(X;Y),}$  the topology of pointwise convergence is metrizable.
• Let ${\displaystyle Y^{X}}$  denote the space of all functions from ${\displaystyle X}$  into ${\displaystyle Y.}$  If ${\displaystyle L(X;Y)}$  is given the topology of pointwise convergence then space of all linear maps (continuous or not) ${\displaystyle X}$  into ${\displaystyle Y}$  is closed in ${\displaystyle Y^{X}}$ .
  • In addition, ${\displaystyle L(X;Y)}$  is dense in the space of all linear maps (continuous or not) ${\displaystyle X}$  into ${\displaystyle Y.}$ 
• Suppose ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex. Any simply bounded subset of ${\displaystyle L(X;Y)}$  is bounded when ${\displaystyle L(X;Y)}$  has the topology of uniform convergence on convex, balanced, bounded, complete subsets of ${\displaystyle X.}$  If in addition ${\displaystyle X}$  is quasi-complete then the families of bounded subsets of ${\displaystyle L(X;Y)}$  are identical for all ${\displaystyle {\mathcal {G}}}$ -topologies on ${\displaystyle L(X;Y)}$  such that ${\displaystyle {\mathcal {G}}}$  is a family of bounded sets covering ${\displaystyle X.}$ ^[13]

Equicontinuous subsets

• The weak-closure of an equicontinuous subset of ${\displaystyle L(X;Y)}$  is equicontinuous.
• If ${\displaystyle Y}$  is locally convex, then the convex balanced hull of an equicontinuous subset of ${\displaystyle L(X;Y)}$  is equicontinuous.
• Let ${\displaystyle X}$  and ${\displaystyle Y}$  be TVSs and assume that (1) ${\displaystyle X}$  is barreled, or else (2) ${\displaystyle X}$  is a Baire space and ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex. Then every simply bounded subset of ${\displaystyle L(X;Y)}$  is equicontinuous.^[11]
• On an equicontinuous subset ${\displaystyle H}$  of ${\displaystyle L(X;Y),}$  the following topologies are identical: (1) topology of pointwise convergence on a total subset of ${\displaystyle X}$ ; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.^[11]

Compact convergence edit

By letting ${\displaystyle {\mathcal {G}}}$  be the set of all compact subsets of ${\displaystyle X,}$  ${\displaystyle L(X;Y)}$  will have the topology of compact convergence or the topology of uniform convergence on compact sets and ${\displaystyle L(X;Y)}$  with this topology is denoted by ${\displaystyle L_{c}(X;Y)}$ .

The topology of compact convergence on ${\displaystyle L(X;Y)}$  has the following properties:

• If ${\displaystyle X}$  is a Fréchet space or a LF-space and if ${\displaystyle Y}$  is a complete locally convex Hausdorff space then ${\displaystyle L_{c}(X;Y)}$  is complete.
• On equicontinuous subsets of ${\displaystyle L(X;Y),}$  the following topologies coincide:
  • The topology of pointwise convergence on a dense subset of ${\displaystyle X,}$ 
  • The topology of pointwise convergence on ${\displaystyle X,}$ 
  • The topology of compact convergence.
  • The topology of precompact convergence.
• If ${\displaystyle X}$  is a Montel space and ${\displaystyle Y}$  is a topological vector space, then ${\displaystyle L_{c}(X;Y)}$  and ${\displaystyle L_{b}(X;Y)}$  have identical topologies.

Topology of bounded convergence edit

By letting ${\displaystyle {\mathcal {G}}}$  be the set of all bounded subsets of ${\displaystyle X,}$  ${\displaystyle L(X;Y)}$  will have the topology of bounded convergence on ${\displaystyle X}$  or the topology of uniform convergence on bounded sets and ${\displaystyle L(X;Y)}$  with this topology is denoted by ${\displaystyle L_{b}(X;Y)}$ .^[6]

The topology of bounded convergence on ${\displaystyle L(X;Y)}$  has the following properties:

• If ${\displaystyle X}$  is a bornological space and if ${\displaystyle Y}$  is a complete locally convex Hausdorff space then ${\displaystyle L_{b}(X;Y)}$  is complete.
• If ${\displaystyle X}$  and ${\displaystyle Y}$  are both normed spaces then the topology on ${\displaystyle L(X;Y)}$  induced by the usual operator norm is identical to the topology on ${\displaystyle L_{b}(X;Y)}$ .^[6]
  • In particular, if ${\displaystyle X}$  is a normed space then the usual norm topology on the continuous dual space ${\displaystyle X^{\prime }}$  is identical to the topology of bounded convergence on ${\displaystyle X^{\prime }}$ .
• Every equicontinuous subset of ${\displaystyle L(X;Y)}$  is bounded in ${\displaystyle L_{b}(X;Y)}$ .

Throughout, we assume that ${\displaystyle X}$  is a TVS.

𝒢-topologies versus polar topologies edit

If ${\displaystyle X}$  is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if ${\displaystyle X}$  is a Hausdorff locally convex space), then a ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle X^{\prime }}$  (as defined in this article) is a polar topology and conversely, every polar topology if a ${\displaystyle {\mathcal {G}}}$ -topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies.

However, if ${\displaystyle X}$  is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in ${\displaystyle X}$ " is stronger than the notion of "${\displaystyle \sigma \left(X,X^{\prime }\right)}$ -bounded in ${\displaystyle X}$ " (i.e. bounded in ${\displaystyle X}$  implies ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ -bounded in ${\displaystyle X}$ ) so that a ${\displaystyle {\mathcal {G}}}$ -topology on ${\displaystyle X^{\prime }}$  (as defined in this article) is not necessarily a polar topology. One important difference is that polar topologies are always locally convex while ${\displaystyle {\mathcal {G}}}$ -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 edit

Suppose that ${\displaystyle X}$  is a TVS whose bounded subsets are the same as its weakly bounded subsets.

Notation: If ${\displaystyle \Delta (Y,X)}$  denotes a polar topology on ${\displaystyle Y}$  then ${\displaystyle Y}$  endowed with this topology will be denoted by ${\displaystyle Y_{\Delta (Y,X)}}$  or simply ${\displaystyle Y_{\Delta }}$  (e.g. for ${\displaystyle \sigma (Y,X)}$  we would have ${\displaystyle \Delta =\sigma }$  so that ${\displaystyle Y_{\sigma (Y,X)}}$  and ${\displaystyle Y_{\sigma }}$  all denote ${\displaystyle Y}$  with endowed with ${\displaystyle \sigma (Y,X)}$ ).

We will let ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  denote the space of separately continuous bilinear maps and ${\displaystyle B(X,Y;Z)}$ denote the space of continuous bilinear maps, where ${\displaystyle X,Y,}$  and ${\displaystyle Z}$  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 ${\displaystyle L(X;Y)}$  we can place a topology on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  and ${\displaystyle B(X,Y;Z)}$ .

Let ${\displaystyle {\mathcal {G}}}$  (respectively, ${\displaystyle {\mathcal {H}}}$ ) be a family of subsets of ${\displaystyle X}$  (respectively, ${\displaystyle Y}$ ) containing at least one non-empty set. Let ${\displaystyle {\mathcal {G}}\times {\mathcal {H}}}$  denote the collection of all sets ${\displaystyle G\times H}$  where ${\displaystyle G\in {\mathcal {G}},}$  ${\displaystyle H\in {\mathcal {H}}.}$  We can place on ${\displaystyle Z^{X\times Y}}$  the ${\displaystyle {\mathcal {G}}\times {\mathcal {H}}}$ -topology, and consequently on any of its subsets, in particular on ${\displaystyle B(X,Y;Z)}$ and on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ . This topology is known as the ${\displaystyle {\mathcal {G}}-{\mathcal {H}}}$ -topology or as the topology of uniform convergence on the products ${\displaystyle G\times H}$  of ${\displaystyle {\mathcal {G}}\times {\mathcal {H}}}$ .

However, as before, this topology is not necessarily compatible with the vector space structure of ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  or of ${\displaystyle B(X,Y;Z)}$ without the additional requirement that for all bilinear maps, ${\displaystyle b}$  in this space (that is, in ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  or in ${\displaystyle B(X,Y;Z)}$ ) and for all ${\displaystyle G\in {\mathcal {G}}}$  and ${\displaystyle H\in {\mathcal {H}},}$  the set ${\displaystyle b(G,H)}$  is bounded in ${\displaystyle X.}$  If both ${\displaystyle {\mathcal {G}}}$  and ${\displaystyle {\mathcal {H}}}$  consist of bounded sets then this requirement is automatically satisfied if we are topologizing ${\displaystyle B(X,Y;Z)}$ but this may not be the case if we are trying to topologize ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ . The ${\displaystyle {\mathcal {G}}-{\mathcal {H}}}$ -topology on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  will be compatible with the vector space structure of ${\displaystyle {\mathcal {B}}(X,Y;Z)}$  if both ${\displaystyle {\mathcal {G}}}$  and ${\displaystyle {\mathcal {H}}}$  consists of bounded sets and any of the following conditions hold:

• ${\displaystyle X}$  and ${\displaystyle Y}$  are barrelled spaces and ${\displaystyle Z}$  is locally convex.
• ${\displaystyle X}$  is a F-space, ${\displaystyle Y}$  is metrizable, and ${\displaystyle Z}$  is Hausdorff, in which case ${\displaystyle {\mathcal {B}}(X,Y;Z)=B(X,Y;Z).}$ 
• ${\displaystyle X,Y,}$  and ${\displaystyle Z}$  are the strong duals of reflexive Fréchet spaces.
• ${\displaystyle X}$  is normed and ${\displaystyle Y}$  and ${\displaystyle Z}$  the strong duals of reflexive Fréchet spaces.

The ε-topology edit

Suppose that ${\displaystyle X,Y,}$  and ${\displaystyle Z}$  are locally convex spaces and let ${\displaystyle {\mathcal {G}}^{\prime }}$  and ${\displaystyle {\mathcal {H}}^{\prime }}$  be the collections of equicontinuous subsets of ${\displaystyle X^{\prime }}$  and ${\displaystyle X^{\prime }}$ , respectively. Then the ${\displaystyle {\mathcal {G}}^{\prime }-{\mathcal {H}}^{\prime }}$ -topology on ${\displaystyle {\mathcal {B}}\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)}^{\prime };Z\right)}$  will be a topological vector space topology. This topology is called the ε-topology and ${\displaystyle {\mathcal {B}}\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)};Z\right)}$  with this topology it is denoted by ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)}^{\prime };Z\right)}$  or simply by ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).}$ 

Part of the importance of this vector space and this topology is that it contains many subspace, such as ${\displaystyle {\mathcal {B}}\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime },Y_{\sigma \left(X^{\prime },X\right)}^{\prime };Z\right),}$  which we denote by ${\displaystyle {\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right).}$  When this subspace is given the subspace topology of ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)}$  it is denoted by ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right).}$ 

In the instance where ${\displaystyle Z}$  is the field of these vector spaces, ${\displaystyle {\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$  is a tensor product of ${\displaystyle X}$  and ${\displaystyle Y.}$  In fact, if ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex Hausdorff spaces then ${\displaystyle {\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$  is vector space-isomorphic to ${\displaystyle L\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime };Y_{\sigma (Y^{\prime },Y)}\right),}$  which is in turn is equal to ${\displaystyle L\left(X_{\tau \left(X^{\prime },X\right)}^{\prime };Y\right).}$ 

These spaces have the following properties:

• If ${\displaystyle X}$  and ${\displaystyle Y}$  are locally convex Hausdorff spaces then ${\displaystyle {\mathcal {B}}_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$  is complete if and only if both ${\displaystyle X}$  and ${\displaystyle Y}$  are complete.
• If ${\displaystyle X}$  and ${\displaystyle Y}$  are both normed (respectively, both Banach) then so is ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ 

• Bornological space – Space where bounded operators are continuous
• Bounded linear operator – Linear transformation between topological vector spacesPages displaying short descriptions of redirect targets
• Dual system
• Dual topology
• List of topologies – List of concrete topologies and topological spaces
• Modes of convergence – Property of a sequence or series
• Operator norm – Measure of the "size" of linear operators
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
• Topologies on the set of operators on a Hilbert space
• Uniform convergence – Mode of convergence of a function sequence
• Uniform space – Topological space with a notion of uniform properties
• Weak topology – Mathematical term
  • Vague topology