Topology of uniform convergence

In mathematics, a linear map is a mapping V → W between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornologies, then one can study the subspace of linear maps that preserve this structure.

Topologies of uniform convergence

Suppose that T is any set and that [math]\displaystyle{ \mathcal{G} }[/math] is a collection of subsets of T directed by inclusion. Suppose in addition that Y is a topological vector space (not necessarily Hausdorff or locally convex) and that [math]\displaystyle{ \mathcal{N} }[/math] is a basis of neighborhoods of 0 in Y. Then the set of all functions from T into Y, [math]\displaystyle{ Y^T }[/math], can be given a unique translation-invariant topology by defining a basis of neighborhoods of 0 in [math]\displaystyle{ Y^T }[/math], to be

[math]\displaystyle{ \mathcal{U}(G, N) = \{f \in Y^T : f(G) \subseteq N\} }[/math]

as G and N range over all [math]\displaystyle{ G \in \mathcal{G} }[/math] and [math]\displaystyle{ N \in \mathcal{N} }[/math]. This topology does not depend on the basis [math]\displaystyle{ \mathcal{N} }[/math] that was chosen and it is known as the topology of uniform convergence on the sets in [math]\displaystyle{ \mathcal{G} }[/math] or as the [math]\displaystyle{ \mathcal{G} }[/math]-topology. In practice, [math]\displaystyle{ \mathcal{G} }[/math] usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, [math]\displaystyle{ \mathcal{G} }[/math] is the collection of compact subsets of T (and T is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of T. A set [math]\displaystyle{ \mathcal{G}_1 }[/math] of [math]\displaystyle{ \mathcal{G} }[/math] is said to be fundamental with respect to [math]\displaystyle{ \mathcal{G} }[/math] if each [math]\displaystyle{ G \in \mathcal{G} }[/math] is a subset of some element in [math]\displaystyle{ \mathcal{G}_1 }[/math]. In this case, the collection [math]\displaystyle{ \mathcal{G} }[/math] can be replaced by [math]\displaystyle{ \mathcal{G}_1 }[/math] without changing the topology on [math]\displaystyle{ Y^T }[/math].

However, the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ Y^T }[/math] is not necessarily compatible with the vector space structure of [math]\displaystyle{ Y^T }[/math] or of any of its vector subspaces (that is, it is not necessarily a topological vector space topology on [math]\displaystyle{ Y^T }[/math]). Suppose that F is a vector subspace [math]\displaystyle{ Y^T }[/math] so that it inherits the subspace topology from [math]\displaystyle{ Y^T }[/math]. Then the [math]\displaystyle{ \mathcal{G} }[/math]-topology on F is compatible with the vector space structure of F if and only if for every [math]\displaystyle{ G \in \mathcal{G} }[/math] and every f ∈ F, f(G) is bounded in Y.

If Y is locally convex then so is the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ Y^T }[/math] and if [math]\displaystyle{ (p_{\alpha}) }[/math] is a family of continuous seminorms generating this topology on Y then the [math]\displaystyle{ \mathcal{G} }[/math]-topology is induced by the following family of seminorms: [math]\displaystyle{ p_{G, \alpha}(f) = \sup_{x \in G} p_{\alpha}(f(x)) }[/math], as G varies over [math]\displaystyle{ \mathcal{G} }[/math] and [math]\displaystyle{ \alpha }[/math] varies over all indices. If Y is Hausdorff and T is a topological space such that [math]\displaystyle{ \bigcup_{G \in \mathcal{G}} G }[/math] is dense in T then the [math]\displaystyle{ \mathcal{G} }[/math]-topology on subspace of [math]\displaystyle{ Y^T }[/math] consisting of all continuous maps is Hausdorff. If the topological space T is also a topological vector space, then the condition that [math]\displaystyle{ \bigcup_{G \in \mathcal{G}} G }[/math] 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.

Let H be a subset of [math]\displaystyle{ Y^T }[/math]. Then H is bounded in the [math]\displaystyle{ \mathcal{G} }[/math]-topology if and only if for every [math]\displaystyle{ G \in \mathcal{G} }[/math], [math]\displaystyle{ \cup_{u \in H} u(G) }[/math] is bounded in Y.

Spaces of continuous linear maps

Throughout this section we will assume that X and Y are topological vector spaces and we will let L(X, Y), denote the vector space of all continuous linear maps from X and Y. If L(X, Y) is given the [math]\displaystyle{ \mathcal{G} }[/math]-topology inherited from [math]\displaystyle{ Y^X }[/math] then this space with this topology is denoted by [math]\displaystyle{ L_{\mathcal{G}}(X, Y) }[/math]. The [math]\displaystyle{ \mathcal{G} }[/math]-topology on L(X, Y) is compatible with the vector space structure of L(X, Y) if and only if for all [math]\displaystyle{ G \in \mathcal{G} }[/math] and all f ∈ L(X, Y) the set f(G) is bounded in Y, which we will assume to be the case for the rest of the article. Note in particular that this is the case if [math]\displaystyle{ \mathcal{G} }[/math] consists of (von-Neumann) bounded subsets of X.

Often, [math]\displaystyle{ \mathcal{G} }[/math] is required to satisfy the following two axioms:

[math]\displaystyle{ \mathcal{G}_1 }[/math]:	If [math]\displaystyle{ G_1, G_2 \in \mathcal{G} }[/math] then there exists a [math]\displaystyle{ G \in \mathcal{G} }[/math] such that [math]\displaystyle{ G_1 \cup G_2 \subseteq G }[/math].
[math]\displaystyle{ \mathcal{G}_2 }[/math]:	If [math]\displaystyle{ G_1 \in \mathcal{G} }[/math] and [math]\displaystyle{ \lambda }[/math] is a scalar then there exists a [math]\displaystyle{ G \in \mathcal{G} }[/math] such that [math]\displaystyle{ \lambda G_1 \subseteq G }[/math].

If [math]\displaystyle{ \mathcal{G} }[/math] is a bornology on X. which is often the case, then these two axioms are satisfied.

Properties

Completeness

For the following theorems, suppose that X is a topological vector space and Y is a locally convex Hausdorff spaces and [math]\displaystyle{ \mathcal{G} }[/math] is a collection of bounded subsets of X that satisfies axioms [math]\displaystyle{ \mathcal{G}_1 }[/math] and [math]\displaystyle{ \mathcal{G}_2 }[/math] and forms a covering of X.

• [math]\displaystyle{ L_{\mathcal{G}}(X, Y) }[/math] is complete if

1. X is locally convex and Hausdorff,
2. Y is complete, and
3. whenever [math]\displaystyle{ u : X \to Y }[/math] is a linear map then u restristed to every set [math]\displaystyle{ G \in \mathcal{G} }[/math] is continuous implies that u is continuous,

• If X is a Mackey space then [math]\displaystyle{ L_{\mathcal{G}}(X, Y) }[/math] is complete if and only if both [math]\displaystyle{ X^*_{\mathcal{G}} }[/math] and Y are complete.
• If X is barrelled then [math]\displaystyle{ L_{\mathcal{G}}(X, Y) }[/math] is Hausdorff and quasi-complete, which means that every closed and bounded set is complete.

Boundedness

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

• H is bounded in [math]\displaystyle{ L_{\mathcal{G}}(X, Y) }[/math],
• For every [math]\displaystyle{ G \in \mathcal{G} }[/math], [math]\displaystyle{ \cup_{u \in H} u(G) }[/math] is bounded in Y,
• For every neighborhood of 0, V, in Y the set [math]\displaystyle{ \cap_{u \in H} u^{-1}(V) }[/math] absorbs every [math]\displaystyle{ G \in \mathcal{G} }[/math].

Furthermore,

• If X and Y are locally convex Hausdorff space and if H is bounded in [math]\displaystyle{ L_{\sigma}(X, Y) }[/math] (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.
• 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 [math]\displaystyle{ \mathcal{G} }[/math]-topologies where [math]\displaystyle{ \mathcal{G} }[/math] is any family of bounded subsets of X covering X.
• If [math]\displaystyle{ \mathcal{G} }[/math] 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 [math]\displaystyle{ \mathcal{G} }[/math]-topology.

Examples

The topology of pointwise convergence L_σ(X, Y)

By letting [math]\displaystyle{ \mathcal{G} }[/math] 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 and L(X, Y) with this topology is denoted by [math]\displaystyle{ L_{\sigma}(X, Y) }[/math]

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

• 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 [math]\displaystyle{ L(X, Y) }[/math] is equicontinuous.
• If A ⊆ X is a countable dense subset of a topological vector space X and if Y is a metrizable topological vector space then [math]\displaystyle{ L_{\sigma}(X, Y) }[/math] is metrizable.
  • So in particular, on every equicontinuous subset of L(X, Y), the topology of pointwise convergence is metrizable.
• Let [math]\displaystyle{ Y^X }[/math] denote the space of all functions from X into Y. If [math]\displaystyle{ F(X, Y) }[/math] is given the topology of pointwise convergence then space of all linear maps (continuous or not) X into Y is closed in [math]\displaystyle{ Y^X }[/math].
  • In addition, L(X, Y) is dense in the space of all linear maps (continuous or not) X into Y.

Compact-convex convergence L_γ(X, Y)

By letting [math]\displaystyle{ \mathcal{G} }[/math] be the set of all compact convex subsets of X, L(X, Y) will have the topology of compact convex convergence or the topology of uniform convergence on compact convex sets L(X, Y) with this topology is denoted by [math]\displaystyle{ L_{\gamma}(X, Y) }[/math].

Compact convergence L_c(X, Y)

By letting [math]\displaystyle{ \mathcal{G} }[/math] 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 [math]\displaystyle{ L_{c}(X, Y) }[/math].

The topology of bounded 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 [math]\displaystyle{ L_{c}(X, Y) }[/math] 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.
• If X is a Montel space and Y is a topological vector space, then [math]\displaystyle{ L_{c}(X, Y) }[/math] and [math]\displaystyle{ L_{b}(X, Y) }[/math] have identical topologies.

Strong dual topology L_b(X, Y)

By letting [math]\displaystyle{ \mathcal{G} }[/math] 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 [math]\displaystyle{ L_{b}(X, Y) }[/math].

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 [math]\displaystyle{ L_{b}(X, Y) }[/math] is complete.
• If X and Y are both normed spaces then [math]\displaystyle{ L_{b}(X, Y) }[/math] is a normed space with the usual operator norm.
• Every equicontinuous subset of L(X, Y) is bounded in [math]\displaystyle{ L_{b}(X, Y) }[/math].

G-topologies on the continuous dual induced by X

The continuous dual space of a topological vector space X over the field [math]\displaystyle{ \mathcal{F} }[/math] (which we will assume to be real or complex numbers) is the vector space [math]\displaystyle{ L(X, \mathcal{F}) }[/math] and is denoted by [math]\displaystyle{ X^* }[/math] and sometimes by [math]\displaystyle{ X' }[/math]. Given [math]\displaystyle{ \mathcal{G} }[/math], a set of subsets of X, we can apply all of the preceding to this space by using [math]\displaystyle{ Y = \mathcal{F} }[/math] and in this case [math]\displaystyle{ X^* }[/math] with this [math]\displaystyle{ \mathcal{G} }[/math]-topology is denoted by [math]\displaystyle{ X^*_{\mathcal{G}} }[/math], so that in particular we have the following basic properties:

• A basis of neighborhoods of 0 for [math]\displaystyle{ X^*_{\mathcal{G}} }[/math] is formed, as [math]\displaystyle{ G }[/math] varies over [math]\displaystyle{ \mathcal{G} }[/math], by the polar sets [math]\displaystyle{ G^\circ := \{x' \in X^* : \sup_{x \in G} |\langle x', x \rangle | \le 1\} }[/math].
  • A filter [math]\displaystyle{ F' }[/math] on [math]\displaystyle{ X^* }[/math] converges to an element [math]\displaystyle{ x' \in X^* }[/math] in the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ X^* }[/math] if [math]\displaystyle{ F' }[/math] uniformly to [math]\displaystyle{ x' }[/math] on each [math]\displaystyle{ G \in \mathcal{G} }[/math].
  • If G ⊆ X is bounded then [math]\displaystyle{ G^\circ }[/math] is absorbing, so [math]\displaystyle{ \mathcal{G} }[/math] usually consists of bounded subsets of X.
• [math]\displaystyle{ X^*_{\mathcal{G}} }[/math] is locally convex,
• If [math]\displaystyle{ \bigcup_{G \in \mathcal{G}} G }[/math] is dense in X then [math]\displaystyle{ X^*_{\mathcal{G}} }[/math] is Hausdorff.
• If [math]\displaystyle{ \bigcup_{G \in \mathcal{G}} G }[/math] covers X then the canonical map from X into [math]\displaystyle{ (X^*_{\mathcal{G}})^* }[/math] is well-defined. That is, for all [math]\displaystyle{ x \in X }[/math] the evaluation functional on [math]\displaystyle{ X^* }[/math] (i.e. [math]\displaystyle{ x' \in X^* \mapsto \langle x', x \rangle }[/math]) is continuous on [math]\displaystyle{ X^*_{\mathcal{G}} }[/math].
  • If in addition [math]\displaystyle{ X^* }[/math] separates points on X then the canonical map of X into [math]\displaystyle{ (X^*_{\mathcal{G}})^* }[/math] is an injection.
• Suppose that X and Y are two topological vector spaces and [math]\displaystyle{ u : E \to F }[/math] is a continuous linear map. Suppose that [math]\displaystyle{ \mathcal{G} }[/math] and [math]\displaystyle{ \mathcal{H} }[/math] are collections of bounded subsets of X and Y, respectively, that both satisfy axioms [math]\displaystyle{ \mathcal{G}_1 }[/math] and [math]\displaystyle{ \mathcal{G}_2 }[/math]. Then [math]\displaystyle{ u }[/math]'s transpose, [math]\displaystyle{ {}^tu : Y^*_{\mathcal{H}} \to X^*_{\mathcal{G}} }[/math] is continuous if for every [math]\displaystyle{ G \in \mathcal{G} }[/math] there is a [math]\displaystyle{ H \in \mathcal{H} }[/math] such that u(G) ⊆ H.
  • In particular, the transpose of [math]\displaystyle{ u }[/math] is continuous if [math]\displaystyle{ X^* }[/math] carries the [math]\displaystyle{ \sigma(X^*, X) }[/math] (respectively, [math]\displaystyle{ \gamma(X^*, X) }[/math], [math]\displaystyle{ c(X^*, X) }[/math], [math]\displaystyle{ b(X^*, X) }[/math]) topology and [math]\displaystyle{ Y^* }[/math] carry any topology stronger than the [math]\displaystyle{ \sigma(Y^*, Y) }[/math] topology (respectively, [math]\displaystyle{ \gamma(Y^*, Y) }[/math], [math]\displaystyle{ c(Y^*, Y) }[/math], [math]\displaystyle{ b(Y^*, Y) }[/math]).
• If X is a locally convex Hausdorff topological vector space over the field [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math] is a collection of bounded subsets of X that satisfies axioms [math]\displaystyle{ \mathcal{G}_1 }[/math] and [math]\displaystyle{ \mathcal{G}_2 }[/math] then the bilinear map [math]\displaystyle{ X \times X^*_{\mathcal{G}} \to \mathcal{F} }[/math] defined by [math]\displaystyle{ (x, x') \mapsto \langle x', x \rangle = x'(x) }[/math] is continuous if and only if X is normable and the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ X^* }[/math] is the strong dual topology [math]\displaystyle{ b(X^*, X) }[/math].
• Suppose that X is a Fréchet space and [math]\displaystyle{ \mathcal{G} }[/math] is a collection of bounded subsets of X that satisfies axioms [math]\displaystyle{ \mathcal{G}_1 }[/math] and [math]\displaystyle{ \mathcal{G}_2 }[/math]. If [math]\displaystyle{ \mathcal{G} }[/math] contains all compact subsets of X then [math]\displaystyle{ X^*_{\mathcal{G}} }[/math] is complete.

Examples

The weak topology σ(X^*, X) or the weak* topology

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

The [math]\displaystyle{ \sigma(X^*, X) }[/math] topology has the following properties:

• Theorem (S. Banach): Suppose that X and Y are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that [math]\displaystyle{ u : X \to Y }[/math] is a continuous linear map. Then [math]\displaystyle{ u }[/math] is surjective if and only if the transpose of [math]\displaystyle{ u }[/math], [math]\displaystyle{ {}^t u : Y^* \to X^* }[/math], is one-to-one and the range of [math]\displaystyle{ {}^t u }[/math] is weakly closed in [math]\displaystyle{ X^*_{\sigma(X^*, X)} }[/math].
• Suppose that X and Y are Fréchet spaces, [math]\displaystyle{ Z }[/math] is a Hausdorff locally convex space and that [math]\displaystyle{ u : X^*_{\sigma} \times Y^*_{\sigma} \to Z^*_{\sigma} }[/math] is a separately-continuous bilinear map. Then [math]\displaystyle{ u : X^*_{b} \times Y^*_{b} \to Z^*_{b} }[/math] 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.
• [math]\displaystyle{ X^*_{\sigma(X^*, X)} }[/math] is normable if and only if X is finite-dimensional.
• When X is infinite-dimensional the [math]\displaystyle{ \sigma(X^*, X) }[/math] topology on [math]\displaystyle{ X^* }[/math] is strictly less fine than the strong dual topology [math]\displaystyle{ b(X^*, X) }[/math].
• The [math]\displaystyle{ \sigma(X^*, X) }[/math]-closure of the convex balanced hull of an equicontinuous subset of [math]\displaystyle{ X^* }[/math] is equicontinuous and [math]\displaystyle{ \sigma(X^*, X) }[/math]-compact.
• Suppose that X is a locally convex Hausdorff space and that [math]\displaystyle{ \hat{X} }[/math] is its completion. If [math]\displaystyle{ X \neq \hat{X} }[/math] then [math]\displaystyle{ \sigma(X^*, \hat{X}) }[/math] is strictly finer than [math]\displaystyle{ \sigma(X^*, X) }[/math].
• Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the [math]\displaystyle{ \sigma(X^*, X) }[/math] topology.

Compact-convex convergence γ(X^*, X)

By letting [math]\displaystyle{ \mathcal{G} }[/math] be the set of all compact convex subsets of X, [math]\displaystyle{ X^* }[/math] will have the topology of compact convex convergence or the topology of uniform convergence on compact convex sets, which is denoted by [math]\displaystyle{ \gamma(X^*, X) }[/math] and [math]\displaystyle{ X^* }[/math] with this topology is denoted by [math]\displaystyle{ X^*_{\gamma} }[/math] or by [math]\displaystyle{ X^*_{\gamma(X^*, X)} }[/math].

• If X is a Fréchet space then the topologies [math]\displaystyle{ \gamma(X^*, X) = c(X^*, X) }[/math].

Compact convergence c(X^*, X)

By letting [math]\displaystyle{ \mathcal{G} }[/math] be the set of all compact subsets of X, [math]\displaystyle{ X^* }[/math] will have the topology of compact convergence or the topology of uniform convergence on compact sets, which is denoted by [math]\displaystyle{ c(X^*, X) }[/math] and [math]\displaystyle{ X^* }[/math] with this topology is denoted by [math]\displaystyle{ X^*_{c} }[/math] or by [math]\displaystyle{ X^*_{c(X^*, X)} }[/math].

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

Precompact convergence

By letting [math]\displaystyle{ \mathcal{G} }[/math] be the set of all precompact subsets of X, [math]\displaystyle{ X^* }[/math] will have the topology of precompact convergence or the topology of uniform convergence on precompact sets.

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

Mackey topology τ(X^*, X)

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

Strong dual topology b(X^*, X)

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

The [math]\displaystyle{ b(X^*, X) }[/math] topology has the following properties:

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

Mackey topology τ(X^*, X^**)

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

• This topology is finer than [math]\displaystyle{ b(X^*, X) }[/math] and hence finer than [math]\displaystyle{ \tau(X^*, X) }[/math].

Other examples

Other [math]\displaystyle{ \mathcal{G} }[/math]-topologies on [math]\displaystyle{ X^* }[/math] include

• The topology of uniform convergence on convex balanced complete bounded subsets of X.
• The topology of uniform convergence on convex balanced infracomplete bounded subsets of X.

G-topologies on X induced by the continuous dual

There is a canonical map from X into [math]\displaystyle{ (X^*_{\sigma})^* }[/math] which maps an element [math]\displaystyle{ x \in X }[/math] to the following map: [math]\displaystyle{ x' \in X^* \mapsto \langle x', x \rangle }[/math]. By using this canonical map we can identify X as being contained in the continuous dual of [math]\displaystyle{ X^*_{\sigma} }[/math] i.e. contained in [math]\displaystyle{ (X^*_{\sigma})^* }[/math]. In fact, this canonical map is onto, which means that [math]\displaystyle{ X = (X^*_{\sigma})^* }[/math] so that we can through this canonical isomorphism think of X as the continuous dual space of [math]\displaystyle{ X^*_{\sigma} }[/math]. Note that it is a common convention that if an equal sign appears between two sets which are clearly not equal, then the equality really means that the sets are isomorphic through some canonical map.

Since we are now regarding X as the continuous dual space of [math]\displaystyle{ X^*_{\sigma} }[/math], we can look at sets of subsets of [math]\displaystyle{ X^*_{\sigma} }[/math], say [math]\displaystyle{ \mathcal{G'} }[/math] and construct a dual space topology on the dual of [math]\displaystyle{ X^*_{\sigma} }[/math], which is X. * A basis of neighborhoods of 0 for [math]\displaystyle{ X_{\mathcal{G'}} }[/math] is formed by the Polar sets [math]\displaystyle{ G'^\circ := \{x \in X : \sup_{x' \in G'} |\langle x', x \rangle | \le 1\} }[/math] as [math]\displaystyle{ G' }[/math] varies over [math]\displaystyle{ \mathcal{G'} }[/math].

Examples

The weak topology σ(X, X^*)

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

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

Convergence on equicontinuous sets ε(X, X^*)

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

• If [math]\displaystyle{ \mathcal{G'} }[/math] was the set of all convex balanced weakly compact equicontinuous subsets of [math]\displaystyle{ X^* }[/math], 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 [math]\displaystyle{ \epsilon(X, X^*) }[/math].

Mackey topology τ(X, X^*)

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

• Suppose that X is a locally convex Hausdorff space. If X is metrizable or barrelled then the initial topology of X is identical to the Mackey topology [math]\displaystyle{ \tau(X, X^*) }[/math].

Bounded convergence b(X, X^*)

By letting [math]\displaystyle{ \mathcal{G} }[/math] be the set of all bounded subsets of X, [math]\displaystyle{ X^* }[/math] will have the topology of bounded convergence or the topology of uniform convergence on bounded sets, which is denoted by [math]\displaystyle{ b(X, X^*) }[/math] and [math]\displaystyle{ X^* }[/math] with this topology is denoted by [math]\displaystyle{ X^*_{b} }[/math] or by [math]\displaystyle{ X^*_{b(X, X^*)} }[/math].

The Mackey–Arens theorem

Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. Any locally convex Hausdorff topological vector space (TVS) topology on X with the property that when X is equipped with this topology has Y as its continuous dual space is said to be compatible with duality between X and Y. If we give X the weak topology [math]\displaystyle{ \sigma(X, Y) }[/math] then [math]\displaystyle{ X_{\sigma(X, Y)} }[/math] is a Hausdorff locally convex topological vector space (TVS) and [math]\displaystyle{ \sigma(X, Y) }[/math] is compatible with duality between X and Y (i.e. [math]\displaystyle{ X_{\sigma(X, Y)}^* = (X_{\sigma(X, Y)})^* = Y }[/math]). 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:

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

1. [math]\displaystyle{ \mathcal{T} }[/math] is identical to a [math]\displaystyle{ \mathcal{G'} }[/math]-topology on X, where [math]\displaystyle{ \mathcal{G'} }[/math] is a covering of [math]\displaystyle{ X^* }[/math] consisting of convex, balanced, [math]\displaystyle{ \sigma(X^*, X) }[/math]-compact sets with the properties that
   1. If [math]\displaystyle{ G_1', G_2' \in \mathcal{G'} }[/math] then there exists a [math]\displaystyle{ G' \in \mathcal{G'} }[/math] such that [math]\displaystyle{ G_1' \cup G_2' \subseteq G' }[/math], and
   2. If [math]\displaystyle{ G_1' \in \mathcal{G'} }[/math] and [math]\displaystyle{ \lambda }[/math] is a scalar then there exists a [math]\displaystyle{ G' \in \mathcal{G'} }[/math] such that [math]\displaystyle{ \lambda G_1' \subseteq G' }[/math].
2. The continuous dual of [math]\displaystyle{ X_{\mathcal{T}} }[/math] is identical to [math]\displaystyle{ X^* }[/math].

And furthermore,

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

G-H-topologies on spaces of bilinear maps

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

Let [math]\displaystyle{ \mathcal{G} }[/math] be a set of subsets of X, [math]\displaystyle{ \mathcal{H} }[/math] be a set of subsets of Y. Let [math]\displaystyle{ \mathcal{G} \times \mathcal{H} }[/math] denote the collection of all sets G × H where [math]\displaystyle{ G \in \mathcal{G} }[/math], [math]\displaystyle{ H \in \mathcal{H} }[/math]. We can place on [math]\displaystyle{ Z^{X \times Y} }[/math] the [math]\displaystyle{ \mathcal{G} \times \mathcal{H} }[/math]-topology, and consequently on any of its subsets, in particular on [math]\displaystyle{ B(X, Y; Z) }[/math] and on [math]\displaystyle{ \mathcal{B}(X, Y; Z) }[/math]. This topology is known as the [math]\displaystyle{ \mathcal{G}-\mathcal{H} }[/math]-topology or as the topology of uniform convergence on the products [math]\displaystyle{ G \times H }[/math] of [math]\displaystyle{ \mathcal{G} \times \mathcal{H} }[/math].

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

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

The ε-topology

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

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

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

These spaces have the following properties:

• If X and Y are locally convex Hausdorff spaces then [math]\displaystyle{ \mathcal{B}_{\epsilon} }[/math][math]\displaystyle{ (X^*_{\sigma}, Y^*_{\sigma}) }[/math] 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 [math]\displaystyle{ \mathcal{B}_{\epsilon} }[/math][math]\displaystyle{ (X^*_{\sigma}, Y^*_{\sigma}) }[/math]

See also

• Bornological space
• Bounded linear operator
• Operator norm
• Uniform convergence
• Uniform space
• Polar topology

References

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co.. pp. xii+144. ISBN 0-7204-0712-5. 
• H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 61–63. ISBN 0-387-05380-8. 
• Trèves, François (1995). Topological Vector Spaces, Distributions and Kernels. Dover Publications. pp. 136–149, 195–201, 240–252, 335–390, 420–433. ISBN 9780486453521. 
• Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.