Weak Continuity Criterion of t for the Continuity of t'

# Weak Continuity Criterion of t for the Continuity of t'

Proposition 1: Let $(E, F)$ and $(G, H)$ be dual pairs, $\mathcal A$ a collection of $\sigma(E, F)$-weakly bounded sets, and let $t : E \to G$ be a linear mapping. Then, if $t$ is weakly continuous then its transpose $t'$ is continuous when $F$ is equipped with the polar topology of $\mathcal A$-convergence and $H$ is equipped with the polar topology of $t(\mathcal A) := \{ t(A) : A \in \mathcal A \}$-convergence. |

*Recall that $t : E \to G$ being weakly continuous simply means that $t$ is continuous when $E$ is equipped with the $\sigma(E, F)$-weak topology and when $G$ is equipped with the $\sigma(G, H)$-weak topology.*

*Also recall that if $t : E \to G$ then $t' : H \to E^*$. But recall that $t$ is weakly continuous if and only if $t'(H) \subseteq F$ (see Weakly Continuous Linear Operators). Thus if $t$ is weakly continuous, then $t'$ maps $H$ to a subset of $F$.*

**Proof:**Let $t$ be weakly continuous so that $t'(H) \subseteq F$ and so that $t' : H \to F$.

- Since $F$ is equipped with the polar topology of $\mathcal A$-convergence, we have that $\{ A^{\circ} : A \in \mathcal A \}$ is a base of neighbourhoods of $o_F$. So for each $A \in \mathcal A$ consider the neighbourhood $A^{\circ}$ of $o_F$. Then

- Since $(E, F)$, $(G, H)$ are dual pairs and $t : E \to G$ is weakly continuous, we have by the proposition on the For Dual Pairs (E, F), (G, H), and a Weakly Continuous Linear Operator t from E to G, (t(A))° = t'-1(A°) page that $t'^{-1}(A^{\circ}) = (t(A))^{\circ}$. Therefore we trivially have that $(t(A))^{\circ} \subseteq t'^{-1}(A^{\circ})$.

- But $(t(A))^{\circ}$ is a neighbourhood of $o_H$ in $H$, when $H$ equipped with the topology of $t(\mathcal A) := \{ t(A) : A \in \mathcal A \}$-convergence.

- Thus $t'$ is continuous when $F$ and $H$ are given the respective topologies above. $\blacksquare$