8.4 The Dual Space
Proposition 8.4.1 (Polarisation, [Proposition 5.5, Fol99]). Let $E$ be a vector space over $\complex$, $\phi \in \hom(E; \complex)$, and $u = \re{\phi}$, then
$u \in \hom(E; \real)$ when $E$ is viewed as a vector space over $\real$.
For any $x \in E$, $\dpb{x, \phi}{E}= \dpb{x, u}{E}- i \dpb{ix, u}{E}$.
Conversely, if $u \in \hom(E; \real)$ and $\phi \in \hom(E; \complex)$ is defined by $\dpb{x, \phi}{E}= \dpb{x, u}{E}- i \dpb{ix, u}{E}$ for all $x \in E$, then $f \in \hom(E; \complex)$.
Proof. (1): Given that $\phi \in \hom(E; \real)$, $u \in \hom(E; \real)$. For any $x \in E$,
so (2) holds.
(2): Since $u \in \hom(E; \real)$, so is $\phi$. On the other hand, for any $x \in E$,
Definition 8.4.2 (Topological Dual). Let $E$ be a TVS over $K \in \RC$, then the topological dual $E^{*}$ of $E$ is the set of all continous linear functionals on $E$.
Definition 8.4.3 (Weak Topology). Let $E$ be a TVS over $K \in \RC$, then the weak topology on $E$ is the initial topology generated by $E^{*}$. The space $E$ equipped with its weak topology is denoted $E_{w}$.