Topologies on spaces of linear maps

• Local convexity: If Y is locally convex then so is the 𝒢-topology on Y^T and if ${\displaystyle (p_{\alpha })}$ is a family of continuous seminorms generating this topology on Y then the 𝒢-topology is induced by the following family of seminorms: ${\displaystyle p_{G,\alpha }(f)=\sup _{x\in G}p_{\alpha }(f(x))}$, as G varies over 𝒢 and ${\displaystyle \alpha }$ varies over all indices.[2]
• Hausdorff: If Y is Hausdorff and T is a topological space such that ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ is dense in T then the 𝒢-topology on the subspace of Y^T consisting of all continuous maps is Hausdorff. If the topological space T is also a topological vector space, then the condition that ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ be dense in T can be replaced by the weaker condition that the linear span of this set be dense in T, in which case we say that this set is total in T.[3]
• Boundedness: Let H be a subset of Y^T. Then H is bounded in the 𝒢-topology if and only if for every G ∈ 𝒢, ${\displaystyle \cup _{u\in H}u(G)}$ is bounded in Y.[2]

Completeness

For the following theorems, suppose that X is a topological vector space and Y is a locally convex Hausdorff spaces and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢₁ and 𝒢₂ and forms a covering of X.

• L_𝒢(X; Y) is complete if

• If X is a Mackey space then L_𝒢(X; Y) is complete if and only if both ${\displaystyle X_{\mathcal {G}}^{\prime }}$ and Y are complete.
• If X is barrelled then L_𝒢(X; Y) is Hausdorff and quasi-complete.
• Let X and Y be TVSs with Y quasi-complete and assume that (1) X is barreled, or else (2) X is a Baire space and X and Y are locally convex. If 𝒢 covers X then every closed equicontinuous subset of L(X; Y) is complete in L_𝒢(X; Y) and L_𝒢(X; Y) is quasi-complete.[4]
• Let X be a bornological space, Y a locally convex space, and 𝒢 a family of bounded subsets of X such that the range of every null sequence in X is contained in some G ∈ 𝒢. If Y is quasi-complete (resp. complete) then so is L_𝒢(X; Y).[5]

Boundedness

Let X and Y be topological vector space and H be a subset of L(X; Y). Then the following are equivalent:[2]

• H is bounded in L_𝒢(X; Y),
• For every G ∈ 𝒢, ${\displaystyle \cup _{u\in H}u(G)}$ is bounded in Y,
• For every neighborhood of 0, V, in Y the set ${\displaystyle \cap _{u\in H}u^{-1}(V)}$ absorbs every G ∈ 𝒢.

Furthermore,

• If X and Y are locally convex Hausdorff space and if H is bounded in ${\displaystyle L_{\sigma }(X;Y)}$ (i.e. pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of X.[6]
• If X and Y are locally convex Hausdorff spaces and if X is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of L(X; Y)' are identical for all 𝒢-topologies where 𝒢 is any family of bounded subsets of X covering X.[6]
• If 𝒢 is any collection of bounded subsets of X whose union is total in X then every equicontinuous subset of L(X; Y) is bounded in the 𝒢-topology.[4]

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

The weak-topology on L(X; Y) has the following properties:

• If X is separable (i.e. has a countable dense subset) and if Y is a metrizable topological vector space then every equicontinuous subset H of ${\displaystyle L_{\sigma }(X,Y)}$ is metrizable; if in addition Y is separable then so is H.[7]
  • So in particular, on every equicontinuous subset of L(X; Y), the topology of pointwise convergence is metrizable.
• Let Y^X denote the space of all functions from X into Y. If ${\displaystyle F(X,Y)}$ is given the topology of pointwise convergence then space of all linear maps (continuous or not) X into Y is closed in Y^X.
  • In addition, L(X; Y) is dense in the space of all linear maps (continuous or not) X into Y.
• Suppose X and Y are locally convex. Any simply bounded subset of L(X; Y) is bounded when L(X; Y) has the topology of uniform convergence on convex, balanced, bounded, complete subsets of X. If in addition X is quasi-complete then the families of bounded subsets of L(X; Y) are identical for all 𝒢-topologies on L(X; Y) such that 𝒢 is a family of bounded sets covering X.[6]

Equicontinuous subsets

• The weak-closure of an equicontinuous subset of L(X; Y) is equicontinuous.
• If Y is locally convex, then the convex balanced hull of an equicontinuous subset of L(X; Y) is equicontinuous.
• Let X and Y be TVSs and assume that (1) X is barreled, or else (2) X is a Baire space and X and Y are locally convex. Then every simply bounded subset of L(X; Y) is equicontinuous.[4]
• On an equicontinuous subset H of L(X; Y), the following topologies are identical: (1) topology of pointwise convergence on a total subset of X; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.[4]

By letting 𝒢 be the set of all compact subsets of X, L(X; Y) will have the topology of compact convergence or the topology of uniform convergence on compact sets and L(X; Y) with this topology is denoted by L_c(X; Y).

The topology of compact convergence on L(X; Y) has the following properties:

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

By letting 𝒢 be the set of all bounded subsets of X, L(X; Y) will have the topology of bounded convergence on X or the topology of uniform convergence on bounded sets and L(X; Y) with this topology is denoted by L_b(X; Y).

The topology of bounded convergence on L(X; Y) has the following properties:

• If X is a bornological space and if Y is a complete locally convex Hausdorff space then L_b(X; Y) is complete.
• If X and Y are both normed spaces then L_b(X; Y) is a normed space with the usual operator norm.
• Every equicontinuous subset of L(X; Y) is bounded in L_b(X; Y).

We have the following basic properties where X and Y are topological vector spaces (not necessarily Hausdorff or locally convex):

• A basis of neighborhoods of 0 for ${\displaystyle X_{\mathcal {G}}^{\prime }}$ is formed, as G varies over 𝒢, by the polar sets ${\displaystyle G^{\circ }:=\{x'\in X^{\prime }:\sup _{x\in G}|\langle x',x\rangle |\leq 1\}}$.
  • A filter ${\displaystyle F'}$ on ${\displaystyle X^{\prime }}$ converges to an element ${\displaystyle x'\in X^{\prime }}$ in the 𝒢-topology on ${\displaystyle X^{\prime }}$ if ${\displaystyle F'}$ uniformly to ${\displaystyle x'}$ on each G ∈ 𝒢.
  • If G ⊆ X is bounded then ${\displaystyle G^{\circ }}$ is absorbing, so 𝒢 usually consists of bounded subsets of X.
• ${\displaystyle X_{\mathcal {G}}^{\prime }}$ is locally convex,
• If ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ is dense in X then ${\displaystyle X_{\mathcal {G}}^{\prime }}$ is Hausdorff.
• If ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ covers X then the canonical map from X into ${\displaystyle (X_{\mathcal {G}}^{\prime })^{\prime }}$ is well-defined. That is, for all ${\displaystyle x\in X}$ the evaluation functional on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle x'\in X^{\prime }\mapsto \langle x',x\rangle }$) is continuous on ${\displaystyle X_{\mathcal {G}}^{\prime }}$.
  • If in addition ${\displaystyle X^{\prime }}$ separates points on X then the canonical map of X into ${\displaystyle (X_{\mathcal {G}}^{\prime })^{\prime }}$ is an injection.
• Suppose that ${\displaystyle u:E\to F}$ is a continuous linear and that 𝒢 and ${\displaystyle {\mathcal {H}}}$ are collections of bounded subsets of X and Y, respectively, that each satisfy axioms 𝒢₁ and 𝒢₂. Then u's transpose, ${\displaystyle {}^{t}u:Y_{\mathcal {H}}^{\prime }\to X_{\mathcal {G}}^{\prime }}$ is continuous if for every G ∈ 𝒢 there is a ${\displaystyle H\in {\mathcal {H}}}$ such that u(G) ⊆ H.[9]
  • In particular, the transpose of u is continuous if ${\displaystyle X^{\prime }}$ carries the ${\displaystyle \sigma (X^{\prime },X)}$ (respectively, ${\displaystyle \gamma (X^{\prime },X)}$, ${\displaystyle c(X^{\prime },X)}$, ${\displaystyle b(X^{\prime },X)}$) topology and ${\displaystyle Y^{\prime }}$ carry any topology stronger than the ${\displaystyle \sigma (Y^{\prime },Y)}$ topology (respectively, ${\displaystyle \gamma (Y^{\prime },Y)}$, ${\displaystyle c(Y^{\prime },Y)}$, ${\displaystyle b(Y^{\prime },Y)}$).
• If X is a locally convex Hausdorff TVS over the field ${\displaystyle {\mathcal {F}}}$ and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢₁ and 𝒢₂ then the bilinear map ${\displaystyle X\times X_{\mathcal {G}}^{\prime }\to {\mathcal {F}}}$ defined by ${\displaystyle (x,x')\mapsto \langle x',x\rangle =x'(x)}$ is continuous if and only if X is normable and the 𝒢-topology on ${\displaystyle X^{\prime }}$ is the strong dual topology ${\displaystyle b(X^{\prime },X)}$.
• Suppose that X is a Fréchet space and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢₁ and 𝒢₂. If 𝒢 contains all compact subsets of X then ${\displaystyle X_{\mathcal {G}}^{\prime }}$ is complete.

${\displaystyle {\mathcal {G}}\subseteq {\mathcal {P}}\left(X\right)}$ ("topology of uniform convergence on ...")	Notation	Name ("topology of...")	Alternative name
finite subsets of X	σ(X', X)	pointwise/simple convergence	weak/weak* topology
precompact subsets		precompact convergence
compact convex subsets of X	γ(X', X)	compact convex convergence
compact subsets of X	c(X', X)	compact convergence
bounded subsets of X	b(X', X)	bounded convergence	strong (dual) topology
convex balanced weakly compact subsets of X	τ(X', X)	Mackey topology
convex balanced complete bounded subsets of X		convex balanced complete bounded convergence
convex balanced infracomplete bounded subsets of X		convex balanced infracomplete bounded convergence

By letting 𝒢 be the set of all finite subsets of X, ${\displaystyle X^{\prime }}$ will have the weak topology on ${\displaystyle X^{\prime }}$ more commonly known as the weak* topology or the topology of pointwise convergence, which is denoted by ${\displaystyle \sigma (X^{\prime },X)}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{\sigma }^{\prime }}$ or by ${\displaystyle X_{\sigma (X^{\prime },X)}^{\prime }}$ if there may be ambiguity.

The ${\displaystyle \sigma (X^{\prime },X)}$ topology has the following properties:

• Alaoglu-Bourbaki Theorem: Every equicontinuous subset of ${\displaystyle X^{\prime }}$ is relatively compact for ${\displaystyle \sigma (X^{\prime },X)}$.[10]
• Theorem (S. Banach): Suppose that X and Y are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that ${\displaystyle u:X\to Y}$ is a continuous linear map. Then ${\displaystyle u}$ is surjective if and only if the transpose of ${\displaystyle u}$, ${\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }}$, is one-to-one and the range of ${\displaystyle {}^{t}u}$ is weakly closed in ${\displaystyle X_{\sigma (X^{\prime },X)}^{\prime }}$.
• Suppose that X and Y are Fréchet spaces, Z is a Hausdorff locally convex space and that ${\displaystyle u:X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime }\to Z_{\sigma }^{\prime }}$ is a separately-continuous bilinear map. Then ${\displaystyle u:X_{b}^{\prime }\times Y_{b}^{\prime }\to Z_{b}^{\prime }}$ is continuous.
  • In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
• ${\displaystyle X_{\sigma (X^{\prime },X)}^{\prime }}$ is normable if and only if X is finite-dimensional.
• When X is infinite-dimensional the ${\displaystyle \sigma (X^{\prime },X)}$ topology on ${\displaystyle X^{\prime }}$ is strictly less fine than the strong dual topology ${\displaystyle b(X^{\prime },X)}$.
• The ${\displaystyle \sigma (X^{\prime },X)}$-closure of the convex balanced hull of an equicontinuous subset of ${\displaystyle X^{\prime }}$ is equicontinuous and ${\displaystyle \sigma (X^{\prime },X)}$-compact.
• Suppose that X is a locally convex Hausdorff space and that ${\displaystyle {\hat {X}}}$ is its completion. If ${\displaystyle X\neq {\hat {X}}}$ then ${\displaystyle \sigma (X^{\prime },{\hat {X}})}$ is strictly finer than ${\displaystyle \sigma (X^{\prime },X)}$.
• Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the ${\displaystyle \sigma (X^{\prime },X)}$ topology.

By letting 𝒢 be the set of all compact convex subsets of X, ${\displaystyle X^{\prime }}$ will have the topology of compact convex convergence or the topology of uniform convergence on compact convex sets, which is denoted by ${\displaystyle \gamma (X^{\prime },X)}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{\gamma }^{\prime }}$ or by ${\displaystyle X_{\gamma (X^{\prime },X)}^{\prime }}$.

• If X is a Fréchet space then the topologies ${\displaystyle \gamma (X^{\prime },X)=c(X^{\prime },X)}$.

By letting 𝒢 be the set of all compact subsets of X, ${\displaystyle X^{\prime }}$ will have the topology of compact convergence or the topology of uniform convergence on compact sets, which is denoted by ${\displaystyle c(X^{\prime },X)}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{c}^{\prime }}$ or by ${\displaystyle X_{c(X^{\prime },X)}^{\prime }}$.

• If X is a Fréchet space or a LF-space then ${\displaystyle c(X^{\prime },X)}$ is complete.
• Suppose that X is a metrizable topological vector space and that ${\displaystyle W'\subseteq X^{\prime }}$. If the intersection of ${\displaystyle W'}$ with every equicontinuous subset of ${\displaystyle X^{\prime }}$ is weakly-open, then ${\displaystyle W'}$ is open in ${\displaystyle c(X^{\prime },X)}$.

By letting 𝒢 be the set of all precompact subsets of X, ${\displaystyle X^{\prime }}$ will have the topology of precompact convergence or the topology of uniform convergence on precompact sets.

• Alaoglu–Bourbaki Theorem: An equicontinuous subset K of ${\displaystyle X^{\prime }}$ has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on K coincides with the ${\displaystyle \sigma (X^{\prime },X)}$ topology.

By letting 𝒢 be the set of all convex balanced weakly compact subsets of X, ${\displaystyle X^{\prime }}$ will have the Mackey topology on ${\displaystyle X^{\prime }}$ or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by ${\displaystyle \tau (X^{\prime },X)}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{\tau (X^{\prime },X)}^{\prime }}$.

By letting 𝒢 be the set of all bounded subsets of X, ${\displaystyle X^{\prime }}$ will have the topology of bounded convergence on X or the topology of uniform convergence on bounded sets or the strong dual topology on ${\displaystyle X^{\prime }}$, which is denoted by ${\displaystyle b(X^{\prime },X)}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{b}^{\prime }}$ or by ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$. Due to its importance, the continuous dual space of ${\displaystyle X_{b}^{\prime }}$ is commonly denoted by ${\displaystyle X^{\prime \prime }}$ (so ${\displaystyle (X_{b}^{\prime })^{\prime }=X^{\prime \prime }}$).

The ${\displaystyle b(X^{\prime },X)}$ topology has the following properties:

• If X is locally convex, then this topology is finer than all other 𝒢-topologies on ${\displaystyle X^{\prime }}$ when considering only 𝒢's whose sets are subsets of X.
• If X is a bornological space (ex: metrizable or LF-space) then ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$is complete.
• If X is a normed space then the strong dual topology on ${\displaystyle X^{\prime }}$ may be defined by the norm ${\displaystyle \|x'\|=\sup _{x\in X,,\|x\|=1}|\langle x',x\rangle |}$, where ${\displaystyle x'\in X^{\prime }}$.[11]
• If X is a LF-space that is the inductive limit of the sequence of space ${\displaystyle X_{k}}$ (for ${\displaystyle k=0,1\dots }$) then ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$ is a Fréchet space if and only if all ${\displaystyle X_{k}}$ are normable.
• If X is a Montel space then
  • ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$ has the Heine–Borel property (i.e. every closed and bounded subset of ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$ is compact in ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$)
  • On bounded subsets of ${\displaystyle X_{b(X^{\prime },X)}^{\prime }}$, the strong and weak topologies coincide (and hence so do all other topologies finer than ${\displaystyle \sigma (X^{\prime },X)}$ and coarser than ${\displaystyle b(X^{\prime },X)}$).
  • Every weakly convergent sequence in ${\displaystyle X^{\prime }}$ is strongly convergent.

By letting ${\displaystyle {\mathcal {G''}}}$ be the set of all convex balanced weakly compact subsets of ${\displaystyle X^{\prime \prime }=(X_{b}^{\prime })^{\prime }}$, ${\displaystyle X^{\prime }}$ will have the Mackey topology on ${\displaystyle X^{\prime }}$ induced by ${\displaystyle X^{\prime \prime }}$' or the topology of uniform convergence on convex balanced weakly compact subsets of ${\displaystyle X^{\prime \prime }}$, which is denoted by ${\displaystyle \tau (X^{\prime },X^{\prime \prime })}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{\tau (X^{\prime },X^{\prime \prime })}^{\prime }}$.

• This topology is finer than ${\displaystyle b(X^{\prime },X)}$ and hence finer than ${\displaystyle \tau (X^{\prime },X)}$.

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all finite subsets of ${\displaystyle X^{\prime }}$, X will have the weak topology or the topology of pointwise convergence on ${\displaystyle X^{\prime }}$, which is denoted by ${\displaystyle \sigma (X,X^{\prime })}$ and X with this topology is denoted by ${\displaystyle X_{\sigma }}$ or by ${\displaystyle X_{\sigma (X,X^{\prime })}}$ if there may be ambiguity.

• Suppose that X and Y are Hausdorff locally convex spaces with X metrizable and that ${\displaystyle u:X\to Y}$ is a linear map. Then ${\displaystyle u:X\to Y}$ is continuous if and only if ${\displaystyle u:\sigma (X,X^{\prime })\to \sigma (Y,Y^{\prime })}$ is continuous. That is, ${\displaystyle u:X\to Y}$ is continuous when X and Y carry their given topologies if and only if ${\displaystyle u}$ is continuous when X and Y carry their weak topologies.

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all equicontinuous subsets ${\displaystyle X^{\prime }}$, X will have the topology of uniform convergence on equicontinuous subsets of ${\displaystyle X^{\prime }}$, which is denoted by ${\displaystyle \epsilon (X,X^{\prime })}$ and X with this topology is denoted by ${\displaystyle X_{\epsilon }}$ or by ${\displaystyle X_{\epsilon (X,X^{\prime })}}$.

• If ${\displaystyle {\mathcal {G'}}}$ was the set of all convex balanced weakly compact equicontinuous subsets of ${\displaystyle X^{\prime }}$, then the same topology would have been induced.
• If X is locally convex and Hausdorff then X's given topology (i.e. the topology that X started with) is exactly ${\displaystyle \epsilon (X,X^{\prime })}$.

Importantly, a set of continuous linear functionals H on a TVS X is equicontinuous if and only if it is contained in the polar of some neighborhood U of 0 in X (i.e. ${\displaystyle H\subseteq U^{\circ }}$). Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of ${\displaystyle X^{\prime }}$ "encode" all information about X's topology (i.e. distinct TVS topologies on X produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of X".

By letting 𝒢 be the set of all bounded subsets of X, ${\displaystyle X^{\prime }}$ will have the topology of bounded convergence or the topology of uniform convergence on bounded sets, which is denoted by ${\displaystyle b(X,X^{\prime })}$ and ${\displaystyle X^{\prime }}$ with this topology is denoted by ${\displaystyle X_{b}^{\prime }}$ or by ${\displaystyle X_{b(X,X^{\prime })}^{\prime }}$.

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all convex balanced weakly compact subsets of ${\displaystyle X^{\prime }}$, X will have the Mackey topology on X or the topology of uniform convergence on convex balanced weakly compact subsets of ${\displaystyle X^{\prime }}$, which is denoted by ${\displaystyle \tau (X,X^{\prime })}$ and X with this topology is denoted by ${\displaystyle X_{\tau }}$ or by ${\displaystyle X_{\tau (X,X^{\prime })}}$.

• Suppose that X is a locally convex Hausdorff space. If X is metrizable or barrelled then X's original topology is identical to the Mackey topology ${\displaystyle \tau (X,X^{\prime })}$.[12]

Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. If 𝜏 is any other locally convex Hausdorff topological vector space topology on X, then we say that 𝜏 is compatible with duality between X and Y if when X is equipped with 𝜏, then it has Y as its continuous dual space. If we give X the weak topology ${\displaystyle \sigma (X,Y)}$ then ${\displaystyle X_{\sigma (X,Y)}}$ is a Hausdorff locally convex topological vector space (TVS) and ${\displaystyle \sigma (X,Y)}$ is compatible with duality between X and Y (i.e. ${\displaystyle X_{\sigma (X,Y)}^{\prime }=(X_{\sigma (X,Y)})^{\prime }=Y}$). We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on X that are compatible with duality between X and Y? The answer to this question is called the Mackey–Arens theorem:[13]

Theorem. Let X be a vector space and let ${\displaystyle {\mathcal {T}}}$ be a locally convex Hausdorff topological vector space topology on X. Let ${\displaystyle X^{\prime }}$ denote the continuous dual space of X and let ${\displaystyle X_{\mathcal {T}}}$ denote X with the topology ${\displaystyle {\mathcal {T}}}$. Then the following are equivalent:

1. ${\displaystyle {\mathcal {T}}}$ is identical to a ${\displaystyle {\mathcal {G'}}}$-topology on X, where ${\displaystyle {\mathcal {G'}}}$ is a covering of ${\displaystyle X^{\prime }}$ consisting of convex, balanced, ${\displaystyle \sigma (X^{\prime },X)}$-compact sets with the properties that
   1. If ${\displaystyle G_{1}',G_{2}'\in {\mathcal {G'}}}$ then there exists a ${\displaystyle G'\in {\mathcal {G'}}}$ such that ${\displaystyle G_{1}'\cup G_{2}'\subseteq G'}$, and
   2. If ${\displaystyle G_{1}'\in {\mathcal {G'}}}$ and ${\displaystyle \lambda }$ is a scalar then there exists a ${\displaystyle G'\in {\mathcal {G'}}}$ such that ${\displaystyle \lambda G_{1}'\subseteq G'}$.
2. The continuous dual of ${\displaystyle X_{\mathcal {T}}}$ is identical to ${\displaystyle X^{\prime }}$.

And furthermore,

1. the topology ${\displaystyle {\mathcal {T}}}$ is identical to the ${\displaystyle \epsilon (X,X^{\prime })}$ topology, that is, to the topology of uniform on convergence on the equicontinuous subsets of ${\displaystyle X^{\prime }}$.
2. the Mackey topology ${\displaystyle \tau (X,X^{\prime })}$ is the finest locally convex Hausdorff TVS topology on X that is compatible with duality between X and ${\displaystyle X_{\mathcal {T}}^{\prime }}$, and
3. the weak topology ${\displaystyle \sigma (X,X^{\prime })}$ is the weakest locally convex Hausdorff TVS topology on X that is compatible with duality between X and ${\displaystyle X_{\mathcal {T}}^{\prime }}$.

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

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

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

These spaces have the following properties:

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