Topologies on spaces of linear maps

𝒢-topologies on spaces of continuous linear maps

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 ${\displaystyle X^{\mathrm{\prime }}}$.

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

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 ${\displaystyle Y.}$

Theorem — Let ${\displaystyle \mathcal{G}}$ be a non-empty collection of bounded subsets of ${\displaystyle X.}$ Then the ${\displaystyle \mathcal{G}}$-topology on ${\displaystyle L(X;Y)}$ is not altered if ${\displaystyle \mathcal{G}}$ is replaced by any of the following collections of (also bounded) subsets of ${\displaystyle X}$:

1. all subsets of all finite unions of sets in ${\displaystyle \mathcal{G}}$;
2. all scalar multiples of all sets in ${\displaystyle \mathcal{G}}$;
3. all finite Minkowski sums of sets in ${\displaystyle \mathcal{G}}$;
4. the balanced hull of every set in ${\displaystyle \mathcal{G}}$;
5. the closure of every set in ${\displaystyle \mathcal{G}}$;

and if ${\displaystyle X}$ and ${\displaystyle Y}$ are locally convex, then we may add to this list:

1. the closed convex balanced hull of every set in ${\displaystyle \mathcal{G}.}$

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 ) 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

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 ${\displaystyle T.}$

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 ${\displaystyle T.}$

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 ${\displaystyle X_{\mathcal{G}}^{\mathrm{\prime }}}$ 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.
• 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 ${\displaystyle L_{\mathcal{G}}(X;Y)}$.

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:

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 ${\displaystyle Y}$;
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. 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 ${\displaystyle X.}$
• 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 ${\displaystyle X.}$

Examples

The topology of pointwise convergence

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; 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 ${\displaystyle H.}$
  • 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 ${\displaystyle X.}$

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.
• 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.

Compact convergence

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

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 ${\displaystyle L_b(X;Y)}$.

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 ${\displaystyle L_b(X;Y)}$.
  • In particular, if ${\displaystyle X}$ is a normed space then the usual norm topology on the continuous dual space ${\displaystyle X^{\mathrm{\prime }}}$ is identical to the topology of bounded convergence on ${\displaystyle X^{\mathrm{\prime }}}$.
• Every equicontinuous subset of ${\displaystyle L(X;Y)}$ is bounded in ${\displaystyle L_b(X;Y)}$.

Polar topologies

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

𝒢-topologies versus polar topologies

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 ${\displaystyle X^{\mathrm{\prime }}}$ (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 "${\displaystyle \sigma \left(X,X^{\mathrm{\prime }}\right)}$-bounded in ${\displaystyle X}$" (i.e. bounded in ${\displaystyle X}$ implies ${\displaystyle \sigma \left(X,X^{\mathrm{\prime }}\right)}$-bounded in ${\displaystyle X}$) so that a ${\displaystyle \mathcal{G}}$-topology on ${\displaystyle X^{\mathrm{\prime }}}$ (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

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

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

𝒢-ℋ topologies on spaces of bilinear maps

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

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

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

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

These spaces have the following properties:

• If ${\displaystyle X}$ and ${\displaystyle Y}$ are locally convex Hausdorff spaces then ${\displaystyle {\mathcal{B}}_{\epsilon }\left(X_{\sigma }^{\mathrm{\prime }},Y_{\sigma }^{\mathrm{\prime }}\right)}$ 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 ${\displaystyle {\mathcal{B}}_{ϵ}\left(X_{\sigma }^{\mathrm{\prime }},Y_{\sigma }^{\mathrm{\prime }}\right)}$