Proposition 18.4.5.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, $\sigma \subset 2^{E}$ be a saturated ideal of $\sigma(E, F)$-bounded sets, $\tau \subset 2^{G}$ be a saturated ideal of $\sigma(G, H)$-bounded sets, then the following are equivalent:
- (1)
$T^{*}$ is continuous with respect to the $\tau$-uniform topology on $H$ and the $\sigma$-uniform topology on $F$.
- (2)
$T(\sigma) \subset \tau$.
Proof. (1) $\Rightarrow$ (2): Let $A \in \sigma$, then there exists $B \in \tau$ such that $T^{*}\phi(A) \subset \ol{B_K(0, 1)}$ for all $\phi \in H$ with $\phi(B) \subset \ol{B_K(0, 1)}$. In which case, $T^{*}(B^{\circ}) \subset A^{\circ}$. Assume without loss of generality that $A$ and $B$ are convex, circled, and closed. By Proposition 18.4.3 applied to $T^{*}$ and the Bipolar theorem, $T(A) \subset B$. Therefore $T(\sigma) \subset \tau$.$\square$
Post a Comment