The Dual Operator and the Weak Topology

Introduction

A continuous linear operator is continuous for every pair of compatible topologies on a dual pair, and in particular it is continuous for the weak topologies; the transpose is continuous for the weak-star topologies, for the weak topologies of the duals, for the Mackey topologies and for the strong topologies. The identification of all these continuities is the statement that the operator and its transpose are the two halves of one morphism of dual pairs. What the weak topologies change is the converse and the sharpness: a linear map is weakly continuous exactly when it is continuous, so no continuity is gained by weakening, and the transpose is a weak-star homeomorphism onto its image exactly when it is injective, which holds when the operator has dense range, while the transpose need not be a topological isomorphism of the strong duals onto its image when the range is not closed.

This article develops the compatibility of an operator with the weak and weak-star topologies. The dual pairs, the weak and weak-star topologies, the Mackey and strong topologies and the bipolar theorem are Duality Theory; the topology of bounded convergence and the strong dual are Operators on a Locally Convex Space; the transpose itself, its algebra properties and its norm are The Dual Operator; the adjoint defined by a pairing and the comparison with the Hilbert-space adjoint are The Dual Pairing and the Adjoint. Nothing analytic and nothing geometric is used.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $E$ and $F$ are Hausdorff locally convex spaces over $\mathbb{K}$, $T \in \mathcal{L}(E, F)$, $T' \in \mathcal{L}(F', E')$ is the transpose, and $\sigma(E, E')$, $\sigma(F', F)$ are the weak and weak-star topologies; the Mackey and strong topologies on a dual are $\tau$ and $\beta$.

Weak Continuity of an Operator

Proposition (the weak topology is initial). On $E$ the weak topology $\sigma(E, E')$ is the initial topology induced by the functionals of $E'$, so a map into $E$ is weakly continuous exactly when all its composites with the functionals of $E'$ are; the continuous functionals for $\sigma(E, E')$ are exactly the elements of $E'$.

Proof. The weak topology is generated by the seminorms $x \mapsto \lvert x'(x)\rvert$, $x' \in E'$, which is the initial topology for the family $E'$; the identification of the dual is the theorem on dual pairs of Duality Theory.

Theorem (an operator is weakly continuous). Every $T \in \mathcal{L}(E, F)$ is continuous for the weak topologies,

$$ T : (E, \sigma(E, E')) \longrightarrow (F, \sigma(F, F')) . $$

Conversely, a linear map $T : E \to F$ is continuous for the given topologies if and only if it is continuous for the weak topologies.

Proof. For $y' \in F'$ the composite $y' \circ T$ lies in $E'$, so it is weakly continuous; by the initial topology property $T$ is weakly continuous. For the converse, weak continuity says $y'\circ T \in (E, \sigma(E, E'))' = E'$ for every $y' \in F'$, so $y' \circ T$ is continuous for the given topology of $E$; a linear map into a locally convex space whose composites with the functionals of the dual are continuous is continuous, because the given topology is the initial topology of the dual family. This is the standard criterion.

Corollary (the weak topology does not change continuity, closed convex sets or bounded sets). The spaces $E$ and $(E, \sigma(E, E'))$ have the same continuous functionals, the same closed convex sets and the same bounded sets; hence an operator is an isomorphism for the given topologies exactly when it is one for the weak topologies, and the weak topologies are the weakest with the same dual.

Proof. The identity $E \to (E, \sigma(E, E'))$ is continuous, and the two spaces have the same dual, so the closed convex sets and the bounded sets coincide; this is Duality Theory.

Continuity of the Transpose for the Weak Topologies

Theorem (weak-star, weak and Mackey continuity). The transpose $T'$ is continuous for the weak-star topologies,

$$ T' : (F', \sigma(F', F)) \longrightarrow (E', \sigma(E', E)) , $$

for the weak topologies $T' : (F', \sigma(F', F'')) \to (E', \sigma(E', E''))$, and for the Mackey topologies $T' : (F', \tau(F', F'')) \to (E', \tau(E', E''))$.

Proof. Weak-star continuity is the characterisation theorem of The Dual Operator: $\langle x, T'y'\rangle = \langle Tx, y'\rangle$, and the continuity of the pairings. For the weak topologies, apply the transpose to the double dual $\iota_{F}T = T''\iota_{E}$; a $\sigma(F', F'')$-continuous functional is evaluation at an element of $F''$, and $T'$ pulls it back to evaluation at the image, so $T'$ is weakly continuous by the initial topology criterion. Mackey continuity follows from the identification of the Mackey topology as the finest compatible topology and the fact that $T'$ is an adjoint of a map of dual pairs; it is quoted from Schaefer–Wolff.

Theorem (injectivity, image and weak-star homeomorphism). The transpose $T'$ is injective if and only if $T$ has dense range; the image is always the annihilator,

$$ T'(F') = (\ker T)^{\perp} \subseteq E' , $$

and $T'$ is a weak-star homeomorphism of $F'$ onto this annihilator exactly when $T$ is surjective.

Proof. $T'y' = 0$ means $y'$ vanishes on $T(E)$, so the kernel of $T'$ is $T(E)^{\perp}$; this is zero exactly when $T(E)$ is dense, by the Hahn–Banach separation theorem in Locally Convex Spaces. For the image, a functional $y' \circ T$ vanishes on $\ker T$, so $T'(F') \subseteq (\ker T)^{\perp}$; conversely a functional $\varphi$ vanishing on $\ker T$ factors through $T(E)$ as $\varphi = \psi \circ T$ with $\psi$ linear on $T(E)$, and $\psi$ extends to $y' \in F'$ by Hahn–Banach, giving $\varphi = T'y'$; hence equality. If $T$ is surjective then for every $y \in F$ there is $x \in E$ with $y = Tx$, so the seminorms $\lvert\langle y, \cdot\rangle\rvert$ defining the weak-star topology of $F'$ and the seminorms $\lvert\langle x, T'\cdot\rangle\rvert$ defining the image topology are the same family, and $T'$ is a homeomorphism onto its image.

Corollary (the transpose of an inclusion). The transpose of an inclusion $i : M \to F$ of a closed subspace is the restriction $i' : F' \to M'$, which is surjective with kernel the annihilator $M^{\perp}$; it is not injective unless $M$ is dense in $F$, and it is a weak-star homeomorphism onto $M'$ exactly when $M = F$.

The Transpose and the Strong Topologies

Proposition (strong continuity). The transpose $T'$ is continuous for the strong topologies $T' : F'_{b} \to E'_{b}$ and for every intermediate pair of topologies of uniform convergence on families of bounded sets carried into one another by $T$.

Proof. The topology of $F'_{b}$ is generated by the seminorms $y' \mapsto \sup_{x\in B}\lvert y'(x)\rvert$ over the bounded $B \subseteq F$; the preimage under $T'$ of the seminorm with $B$ is the seminorm of $E'_{b}$ with the bounded set $T^{-1}(B)$, which is bounded; hence continuity. The statement for intermediate families is the same computation with the family transported by $T$.

Proposition (the transpose need not be an isomorphism of the duals). Strong continuity of the transpose does not make it a topological isomorphism onto its image: for the inclusion $i : c_{0} \to \ell^{\infty}$ the transpose is the restriction $(\ell^{\infty})' \to (c_{0})' = \ell^{1}$, which is surjective with the nonzero kernel $c_{0}^{\perp}$, so $i'$ is not injective even though $i$ has closed range; the same phenomenon occurs for every proper closed subspace whose annihilator is nonzero.

Proof. The transpose of an inclusion $i : M \to F$ is the restriction $F' \to M'$, which is surjective when $M$ is closed, by Hahn–Banach, with kernel the annihilator $M^{\perp}$; for $M = c_{0} \subseteq \ell^{\infty}$ the annihilator consists of the nonzero functionals vanishing on the null sequences, which exist by Hahn–Banach, so $i'$ is not injective. Injectivity would require $M$ dense, which a proper closed subspace is not.

Summary

A continuous operator $T \in \mathcal{L}(E, F)$ is continuous for the weak topologies, and conversely a linear map is continuous exactly when it is weakly continuous; the weak topology has the same dual, the same closed convex sets and the same bounded sets as the given topology, which is why the operator and its continuity are unchanged by it. The transpose $T'$ is continuous for the weak-star topologies — which characterises transposes — and also for the weak topologies of the duals, for the Mackey topologies and for the strong topologies. It is injective exactly when $T$ has dense range, its image is always the annihilator $(\ker T)^{\perp}$, and it is a weak-star homeomorphism onto that annihilator exactly when $T$ is surjective; the transpose of an inclusion is the restriction on the duals, which is surjective with kernel the annihilator and is not injective for a proper closed subspace, as $c_{0} \subseteq \ell^{\infty}$ shows. The weak and weak-star topologies therefore see the operator, its transpose and their duality relations, and add no continuity beyond that of the operator itself.

Summary of Notation

Symbol	Meaning
$\sigma(E, E')$	Weak topology on $E$
$\sigma(E', E)$	Weak-star topology on $E'$
$\tau$, $\beta$	Mackey and strong topologies
$T' : F' \to E'$	Transpose, $\langle Tx, y'\rangle = \langle x, T'y'\rangle$
$T'' = (T')'$, $T''\iota_{E} = \iota_{F}T$	Double dual and naturality
$T'(F') = (\ker T)^{\perp}$	Image of the transpose, the annihilator
dense range	$T(E)$ dense in $F$; equivalent to injectivity of $T'$
surjective $T$	$T'$ a weak-star homeomorphism onto $(\ker T)^{\perp}$