Proposition 18.4.3.label Let $\dpn{E, F}{\lambda}$ and $\dpn{G, H}{\mu}$ be dualities over $K \in \RC$, $T: E \to G$ be a $\sigma(E, F)$-$\sigma(G, H)$ continuous linear map, $A \subset E$, and $B \subset G$, then:
- (1)
$T(A)^{\circ} = (T^{*})^{-1}(A^{\circ})$.
- (2)
If $T(A) \subset B$, then $T^{*}(B^{\circ}) \subset A^{\circ}$.
Proof. (1):
\begin{align*}T(A)^{\circ}&= \bracsn{\phi \in H| \text{Re}\dpn{Tx, \phi}{\mu} \le 1 \forall x \in A}\\&= \bracsn{\phi \in H| \text{Re}\dpn{x, T^*\phi}{\lambda} \le 1 \forall x \in A}= (T^{*})^{-1}(A^{\circ})\end{align*}
(2):
\begin{align*}T^{*}(B^{\circ})&= T^{*}(\bracs{\phi \in H| \text{Re}\dpn{y, \phi}{\mu} \le 1 \forall y \in B}) \\&\subset T^{*}(\bracs{\phi \in H| \text{Re}\dpn{y, \phi}{\mu} \le 1 \forall y \in T(A)}) \\&= T^{*}(\bracs{\phi \in H| \text{Re}\dpn{Tx, \phi}{\mu} \le 1 \forall x \in A})\\&= T^{*}(\bracs{\phi \in H| \text{Re}\dpn{x, T^*\phi}{\lambda} \le 1 \forall x \in A}) \subset A^{\circ}\end{align*}
$\square$
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