Topological dual - Imported Knowlpedia Content

Definition

Let $E$ be a topological vector space over $\mathbb K=\mathbb R$ or $\mathbb C$ . Its topological dual, continuous dual, or continuous linear dual is

E'=\{\varphi:E\to\mathbb K:\varphi\text{ is a continuous linear map}\}.

With pointwise addition and scalar multiplication, $E'$ is a vector space. Evaluation gives the canonical duality pairing

\langle \varphi,x\rangle=\varphi(x).

The topological dual is generally smaller than the algebraic dual. Moreover, the notation $E'$ specifies a set of functionals, not a topology on that set; weak, weak-star, strong, and operator-norm topologies are additional choices determined by the setting.

Continuity criteria

For a normed space $E$ , a linear functional is continuous exactly when it is bounded on the unit ball, and

\lVert\varphi\rVert=\sup_{\lVert x\rVert\leq1}|\varphi(x)|

defines the dual norm. For a locally convex space whose topology is generated by seminorms, continuity means that $|\varphi(x)|$ is bounded by a constant times the maximum of finitely many generating seminorms.

Natural dual topologies

The weak-star topology $\sigma(E',E)$ is pointwise convergence on $E$ . The strong dual topology is uniform convergence on bounded subsets of $E$ . When $E$ is normed, the dual norm topology is uniform convergence on its unit ball. These topologies need not coincide, so statements about convergence or continuity on $E'$ must name the chosen topology.

Separation and examples

For a Hausdorff locally convex space, the Hahn–Banach theorem ensures that $E'$ separates points: if $x\neq0$ , some $\varphi\in E'$ satisfies $\varphi(x)\neq0$ . For $1\leq p<\infty$ , the familiar $L^q$ -pairing describes the dual of $L^p$ under standard measure-theoretic hypotheses. By contrast, the continuous dual can be trivial for non-locally-convex topological vector spaces.

References

Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 3.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier book record. Relevant: Chapter 13.