Continuous anti-dual

For a complex locally convex space $E$ , its continuous anti-dual is

E^\times=\{\lambda:E\to\mathbb C:\lambda\text{ is continuous and conjugate-linear}\}.

Conjugate-linearity means $\lambda(av+bw)=\overline a\lambda(v)+\overline b\lambda(w)$ . Pointwise addition and scalar multiplication make $E^\times$ a complex vector space. Its strong anti-dual topology is uniform convergence on bounded subsets, with seminorms

q_B(\lambda)=\sup_{w\in B}|\lambda(w)|.

This is the anti-linear version of the strong dual topology.

Hilbert-space embedding

Use an inner product linear in its first argument. If $E\hookrightarrow H$ is continuous and dense, then

j:H\longrightarrow E^\times,\qquad j(v)(w)=\langle v,w\rangle_H

is a continuous linear injection. Indeed, $q_B(j(v))\leq\|v\|\sup_{w\in B}\|w\|$ , and density proves injectivity.

Relation to the linear dual

The map $\lambda\mapsto\overline\lambda$ , with $\overline\lambda(w)=\overline{\lambda(w)}$ , identifies the anti-dual conjugate-linearly with the continuous linear dual. Thus linear and anti-linear distribution conventions differ by complex conjugation. The anti-dual convention makes $j$ linear without choosing a conjugation on $H$ .