Strong dual of a locally convex space - Imported Knowlpedia Content

Definition

Let $E$ be a Hausdorff locally convex space with topological dual $E'$ . The strong dual $E'_\beta$ , also written $E'_b$ or $(E',\beta(E',E))$ , is $E'$ equipped with the locally convex topology of uniform convergence on bounded subsets of $E$ . It is generated by the seminorms

p_B(\varphi)=\sup_{x\in B}\lvert\varphi(x)\rvert,

where $B$ ranges over bounded subsets of $E$ . Each $p_B$ is finite because a continuous linear functional sends bounded sets to bounded sets.

Relation to other dual topologies

The weak-star topology $\sigma(E',E)$ is uniform convergence only on finite subsets of $E$ , so it is no finer than the strong topology. If $E$ is normed, every bounded set is contained in a scalar multiple of its unit ball; consequently the strong dual topology is exactly the usual dual-norm topology. For a general locally convex space, no single bounded set need control all others.

Boundedness governs the topology

Two locally convex topologies on the same vector space that have the same bounded subsets induce the same strong topology on their common continuous dual. This dependence on bounded sets makes strong duals natural for distributions. In particular, $\mathcal S'(\mathbb R^n)$ is commonly given the strong dual topology of the Schwartz space, which is finer than pointwise convergence on test functions Trèves, Chapters 19 and 33.

Conventions and cautions

The word strong here concerns uniform convergence on bounded subsets of a locally convex space. It should not be confused with the strong operator topology on an operator algebra. Authors may write $E'_b$ , $E'_\beta$ , or simply $E'$ when the strong topology is understood; the last notation is ambiguous unless the surrounding text declares its dual-topology convention.

References

François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967; Dover reprint, 2006. Dover publisher record. Relevant: Chapters 19 and 33 on dual topologies and strong duals.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter IV on topologies of uniform convergence.