Proposition 18.4.2.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$ and $T \in \hom(E; G)$, then the following are equivalent:

1. (1)

   $T$ is $\sigma(E, F)$-$\sigma(G, H)$ continuous.

2. (2)

   $T^{*}(H) \subset F$.

If the above holds, then

1. (3)

   $T^{*}|_{H}$ is $\sigma(H, G)$-$\sigma(F, E)$ continuous.

2. (4)

   $T^{**}= T$.

and the restriction of $T^{*}$ to $H$ is the adjoint of $T$ with respect to $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$.