Operators on a Locally Convex Space

Introduction

The continuous operators of a locally convex space form a vector space, and on it there are several natural locally convex topologies, distinguished by the families of sets on which the operators are required to converge uniformly. The finest of the common ones is the topology of bounded convergence, uniform convergence on the bounded subsets of the domain; its neighbourhoods of zero are the sets of operators carrying a bounded set into a prescribed neighbourhood of zero, and with it the operator space becomes a locally convex space whose complete subsets are the closed, bounded, equicontinuous families. The same construction applied to the functionals is the strong topology on the dual, so the strong dual is the case $F = \mathbb{K}$ of the operator space.

This article develops that topology on $\mathcal{L}(E, F)$: its seminorms and neighbourhoods, the resulting uniform structure and the completeness inherited from the target, the identification of its bounded subsets, the comparison with the topologies of pointwise and of compact convergence, and the standard examples. The duality-theoretic content — the strong dual, the polar calculus and the Banach–Alaoglu theorem — is Duality Theory; the transpose and its continuity for these topologies are The Dual Operator and The Dual Operator and the Weak Topology; the operator algebra and the completeness of the normed case are Bounded Operators and the Operator Norm. The space of continuous operators itself, the operator norm and the bornological criterion are Bounded Operators on a Topological Vector Space. Nothing analytic and nothing geometric is used; the spectral theory of these operators is Analysis on Linear Spaces in Part III.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $E$ and $F$ are Hausdorff locally convex spaces over $\mathbb{K}$ with duals $E'$ and $F'$, $\mathcal{L}(E, F)$ is the space of continuous linear maps, $\mathcal{B}$ is the family of bounded subsets of $E$, and $q$ runs over the continuous seminorms of $F$. For a bounded set $B \subseteq E$ and a neighbourhood $V$ of $0$ in $F$ the set of operators carrying $B$ into $V$ is written $N(B, V) = \{T \in \mathcal{L}(E, F) : T(B) \subseteq V\}$.

The Topology of Bounded Convergence

Definition. The topology of bounded convergence on $\mathcal{L}(E, F)$ is the locally convex topology generated by the seminorms

$$ q_{B}(T) = \sup_{x \in B} q(Tx) \qquad (B \in \mathcal{B}, \ q \text{ a continuous seminorm on } F), $$

that is, the topology of uniform convergence on the bounded subsets of $E$. The space $\mathcal{L}(E, F)$ with this topology is written $\mathcal{L}_{b}(E, F)$; the special case $\mathcal{L}_{b}(E, \mathbb{K}) = E'_{b}$ is the strong dual.

Proposition (neighbourhoods and linear structure). The sets $N(B, V)$, for $B \in \mathcal{B}$ and $V$ a convex balanced neighbourhood of $0$ in $F$, form a fundamental system of convex balanced neighbourhoods of $0$ for the topology of bounded convergence; the operations $(S, T) \mapsto S + T$ and $(\lambda, T) \mapsto \lambda T$ are continuous, so $\mathcal{L}_{b}(E, F)$ is a locally convex space, and composition $\mathcal{L}(F, G) \times \mathcal{L}(E, F) \to \mathcal{L}(E, G)$ is separately continuous.

Proof. $N(B, V)$ is the intersection of the conditions $q(Tx) \leq 1$ running over $x \in B$ and over the finitely many seminorms defining $V$, so it is a convex balanced neighbourhood of $0$, and every such neighbourhood contains one of the form $N(B, V)$. Additivity is clear; for the scalar multiplication, $\lambda T \in N(B, V)$ whenever $T \in N(B, (\lambda^{-1})V)$ and $\lambda \neq 0$. Separate continuity of composition is the observation that $S \mapsto ST$ carries $N(S(B), V)$ back to $N(B, S^{-1}(V))$ reasoning on the range, and $T \mapsto ST$ carries $N(B, V)$ into $N(B, S(V))$.

Proposition (comparison of topologies). The topology of bounded convergence is finer than the topology of compact convergence (uniform convergence on compact sets) and than the topology of pointwise convergence (the strong operator topology). All three coincide when every bounded subset of $E$ is relatively compact, in particular on a Montel space.

Proof. A compact set is bounded, and a finite set is compact, so the families of seminorms are nested; if every bounded set is relatively compact then the sup over a bounded set equals the sup over its compact closure, and the three families define the same topology. The Montel case is Locally Convex Spaces.

Proposition (the normed case). Let $X$ be a normed space and $F$ a locally convex space. Then the topology of bounded convergence on $\mathcal{L}(X, F)$ is generated by the seminorms $T \mapsto \sup_{\lVert x\rVert \leq 1} q(Tx)$; when $F$ is normed this is the topology of the operator norm, so $\mathcal{L}_{b}(X, Y) = B(X, Y)$ for normed $X, Y$.

Proof. Bounded subsets of a normed space are contained in a multiple of the unit ball, and the supremum over a multiple is a scalar multiple of the supremum over the unit ball, so the single seminorm with $B$ the unit ball generates the topology; when $F$ is normed with norm $\lVert\cdot\rVert$, that seminorm is $T \mapsto \lVert T\rVert$.

Completeness and Bounded Sets

Theorem (completeness). If $F$ is quasi-complete then $\mathcal{L}_{b}(E, F)$ is quasi-complete; if $F$ is complete then $\mathcal{L}_{b}(E, F)$ is complete. In particular the strong dual $E'_{b}$ of a Fréchet space is complete.

Proof. Let $(T_{\alpha})$ be a Cauchy net. For each $x$ the net $(T_{\alpha}x)$ is Cauchy in $F$, because $\{x\}$ is bounded; if $F$ is complete it has a limit $Tx$, and $T$ is linear. For continuity, let $V$ be a closed convex balanced neighbourhood of $0$ in $F$ and choose $W$ with $W + W \subseteq V$; by the Cauchy condition on the bounded set $\{x_{\lambda}\}$ of a net $x_{\lambda} \to 0$ there is $\alpha_{0}$ with $T_{\beta}x_{\lambda} - T_{\alpha_{0}}x_{\lambda} \in W$ for all $\beta \geq \alpha_{0}$, and passing to the limit gives $Tx_{\lambda} - T_{\alpha_{0}}x_{\lambda} \in V$; since $T_{\alpha_{0}}x_{\lambda} \to 0$, one has $Tx_{\lambda} \in V + V \subseteq 2V$ eventually, so $Tx_{\lambda} \to 0$ and $T$ is continuous. The quasi-complete statement is the same argument with the closed bounded subsets of $F$, which are complete by hypothesis.

Theorem (bounded sets are equicontinuous). Let $E$ be barrelled and let $H \subseteq \mathcal{L}(E, F)$ be a set of operators that is bounded for the topology of bounded convergence, that is, $H(B)$ is bounded in $F$ for every bounded $B \subseteq E$. Then $H$ is equicontinuous.

Proof. This is the uniform boundedness principle (Banach–Steinhaus) in the form proved in Locally Convex Spaces: on a barrelled space a family of continuous linear maps that is bounded on each bounded set — equivalently, pointwise bounded on each bounded set — is equicontinuous, and the barrels $\bigcap_{T \in H} T^{-1}(V)$ are neighbourhoods of $0$. The equivalence of the two boundedness conditions is the definition of boundedness in $\mathcal{L}_{b}(E, F)$.

Corollary (closed bounded equicontinuous sets are compact). If $E$ is barrelled and $F$ is a Fréchet space, every closed, bounded, equicontinuous subset of $\mathcal{L}_{b}(E, F)$ is compact. In particular the closed bounded sets of the strong dual of a barrelled space are weak-star compact, which is the Banach–Alaoglu theorem.

Proof. An equicontinuous family on a barrelled domain is a subset of a product of compact sets, by Banach–Alaoglu applied to the polars of a neighbourhood of $0$ in $E$, and closedness makes it compact; the identification of the closed bounded sets of $E'_{b}$ with the weak-star closed equicontinuous ones is Duality Theory.

Examples

Example (the strong dual). For $F = \mathbb{K}$ the topology of bounded convergence on $\mathcal{L}(E, \mathbb{K}) = E'$ is the strong topology $\beta(E', E)$ of uniform convergence on the bounded sets, and $\mathcal{L}_{b}(E, \mathbb{K}) = E'_{b}$. Its bounded sets are the weakly bounded sets, which are the strongly bounded sets, by Duality Theory.

Example (the operators of a normed space). For normed $X, Y$ the topology of bounded convergence is the operator-norm topology, and $\mathcal{L}_{b}(X, Y) = B(X, Y)$ is a normed space, complete when $Y$ is complete; this is the identification of Bounded Operators and the Operator Norm.

Example (pointwise against bounded convergence). On $E = \ell^{2}$ the operators $T_{n}x = \langle x, e_{n}\rangle e_{1}$ satisfy $T_{n}x \to 0$ for every $x$, but $\lVert T_{n}\rVert = 1$ for every $n$, so $T_{n} \to 0$ for the topology of pointwise convergence and not for the topology of bounded convergence, which here is the norm topology. The two topologies therefore differ, while having the same bounded sets, since on the barrelled space $\ell^{2}$ a pointwise bounded family of operators is equicontinuous and hence bounded for the topology of bounded convergence.

Example (the transpose). For $T \in \mathcal{L}(E, F)$ the transpose $T' : F'_{b} \to E'_{b}$ is continuous for the strong topologies: the preimage of $N(B, V)$ is $N(T(B), V^{\circ\circ})$ with $T(B)$ bounded, which is the definition of continuity. The transpose of The Dual Operator is thus a continuous map of the strong duals, and its continuity for the weak and weak-star topologies is The Dual Operator and the Weak Topology.

Summary

On the space $\mathcal{L}(E, F)$ of continuous operators of locally convex spaces the topology of bounded convergence is the topology of uniform convergence on the bounded subsets of $E$, generated by the seminorms $q_{B}(T) = \sup_{x \in B}q(Tx)$; its convex balanced neighbourhoods of $0$ are the sets $N(B, V)$ of operators carrying a bounded set $B$ into a neighbourhood $V$ of $0$, and it makes $\mathcal{L}_{b}(E, F)$ a locally convex space on which composition is separately continuous. It is finer than the topologies of compact convergence and of pointwise convergence, which coincide with it when the bounded sets are relatively compact, and in the normed case it is the operator-norm topology. The space $\mathcal{L}_{b}(E, F)$ is quasi-complete when $F$ is quasi-complete and complete when $F$ is complete, so the strong dual $E'_{b} = \mathcal{L}_{b}(E, \mathbb{K})$ is complete for a Fréchet space; the bounded subsets of $\mathcal{L}_{b}(E, F)$ are, on a barrelled domain, exactly the equicontinuous families, by the uniform boundedness principle, and a closed bounded equicontinuous subset of a strong dual is weak-star compact, which is Banach–Alaoglu. The transpose is continuous for the strong topologies.

Summary of Notation

Symbol	Meaning
$\mathcal{L}(E, F)$	Continuous linear maps of locally convex spaces
$\mathcal{L}_{b}(E, F)$	$\mathcal{L}(E, F)$ with the topology of bounded convergence
$\mathcal{B}$	Family of bounded subsets of $E$
$q_{B}(T) = \sup_{x \in B}q(Tx)$	Generating seminorm of the bounded-convergence topology
$N(B, V) = \{T : T(B) \subseteq V\}$	Basic convex balanced neighbourhood of $0$
$E'_{b}$	Strong dual, $\mathcal{L}_{b}(E, \mathbb{K})$
compact convergence, pointwise convergence	Coarser topologies on $\mathcal{L}(E, F)$
equicontinuous	Family whose inverse images of a neighbourhood form a neighbourhood
barrelled	Locally convex space on which every barrel is a neighbourhood