Corollary 18.4.4.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, then:

1. (1)

   $\ker(T^{*}) = T(E)^{\perp} = \bracs{\phi \in H| \dpn{y, \phi}{\mu} = 0 \forall y \in T(E)}$.

2. (2)

   $T^{*}$ is injective if and only if $T(E)$ is $\sigma(G, H)$-dense in $G$.