Topologies on spaces of linear maps

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

$${\displaystyle {\mathcal {U}}(G,N):=\{f\in F:f(G)\subseteq N\}.}$$

The family

$${\displaystyle \{{\mathcal {U}}(G,N):G\in {\mathcal {G}},N\in {\mathcal {N}}\}}$$

forms a neighborhood basis[1] at the origin for a unique translation-invariant topology on ${\displaystyle F,}$ where this topology is not necessarily a vector topology (that is, it might not make ${\displaystyle F}$ into a TVS). This topology does not depend on the neighborhood basis ${\displaystyle {\mathcal {N}}}$ that was chosen and it is known as the topology of uniform convergence on the sets in ${\displaystyle {\mathcal {G}}}$ or as the ${\displaystyle {\mathcal {G}}}$-topology.[2] However, this name is frequently changed according to the types of sets that make up ${\displaystyle {\mathcal {G}}}$ (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details[3]).

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]

$${\displaystyle {\mathcal {U}}(G,N)-{\mathcal {U}}(G,N)\subseteq {\mathcal {U}}(G,N-N)}$$

and similarly[5]

$${\displaystyle {\mathcal {U}}(G,M)+{\mathcal {U}}(G,N)\subseteq {\mathcal {U}}(G,M+N).}$$

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

$${\displaystyle {\mathcal {U}}(G\cup H,M\cap N)\subseteq {\mathcal {U}}(G,M)\cap {\mathcal {U}}(H,N)}$$

which implies:

• 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]

$${\displaystyle {\mathcal {U}}\left(\bigcup _{S\in {\mathcal {S}}}S,N\right)=\bigcap _{S\in {\mathcal {S}}}{\mathcal {U}}(S,N)\qquad {\text{ and }}\qquad {\mathcal {U}}\left(G,\bigcap _{M\in {\mathcal {M}}}M\right)=\bigcap _{M\in {\mathcal {M}}}{\mathcal {U}}(G,M).}$$

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

$${\displaystyle {\mathcal {W}}(G,U)~:=~\left\{(u,v)\in Y^{T}\times Y^{T}~:~(u(g),v(g))\in U\;{\text{ for every }}g\in G\right\}.}$$

Given ${\displaystyle G\subseteq T,}$ the family of all sets ${\displaystyle {\mathcal {W}}(G,U)}$ as ${\displaystyle U}$ ranges over any fundamental system of entourages of ${\displaystyle Y}$ forms a fundamental system of entourages for a uniform structure on ${\displaystyle Y^{T}}$ called the uniformity of uniform converges on ${\displaystyle G}$ or simply the ${\displaystyle G}$-convergence uniform structure.[7] The ${\displaystyle {\mathcal {G}}}$-convergence uniform structure is the least upper bound of all ${\displaystyle G}$-convergence uniform structures as ${\displaystyle G\in {\mathcal {G}}}$ ranges over ${\displaystyle {\mathcal {G}}.}$[7]

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]

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:

$${\displaystyle p_{G,i}(f):=\sup _{x\in G}p_{i}(f(x)),}$$

as ${\displaystyle G}$ varies over ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle i}$ varies over ${\displaystyle I}$.[8]

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]

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]

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.

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]

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]

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.

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 ${\displaystyle L_{b}(X;Y)}$.

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.

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 ${\displaystyle \sigma (Y,X)}$).

>${\displaystyle {\mathcal {G}}\subseteq \wp (X)}$ ("topology of uniform convergence on ...")	Notation	Name ("topology of...")	Alternative name
finite subsets of ${\displaystyle X}$	${\displaystyle \sigma (Y,X)}$ ${\displaystyle s(Y,X)}$	pointwise/simple convergence	weak/weak* topology
${\displaystyle \sigma (X,Y)}$-compact disks	${\displaystyle \tau (Y,X)}$		Mackey topology
${\displaystyle \sigma (X,Y)}$-compact convex subsets	${\displaystyle \gamma (Y,X)}$	compact convex convergence
${\displaystyle \sigma (X,Y)}$-compact subsets (or balanced ${\displaystyle \sigma (X,Y)}$-compact subsets)	${\displaystyle c(Y,X)}$	compact convergence
${\displaystyle \sigma (X,Y)}$-bounded subsets	${\displaystyle b(Y,X)}$ ${\displaystyle \beta (Y,X)}$	bounded convergence	strong topology

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 ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$