# Is a sequentially continuous map $f:E'\to E'$ continuous?

I have read that for a separable complete locally convex space $E$, any sequentially continuous linear map $f:E’\to \mathbb{K}$ is continuous (where $E’$ is equipped with the weak*-topology).

Is there any similar result for a linear map $f:E’\to E’$?

#### Solutions Collecting From Web of "Is a sequentially continuous map $f:E'\to E'$ continuous?"

Let $E$ be a separable complete locally convex space, $f:(E’,w^*)\to (E’,w^*)$ a sequentially continuous linear map and $ev_x:E’\ni x’\mapsto ev_x(x’)=x'(x)\in \mathbb K$ the evaluation map. Then:

\begin{align}
This shows that $ev_x\circ f:(E’,w^*)\to \mathbb K$ is sequentially continuous for all $x\in E$ and thus, by the result in the post, $ev_x\circ f:(E’,w^*)\to \mathbb K$ is continuous for all $x\in E$. It follows that the map $f:(E’,w^*)\to (E’,w^*)$ is continuous.
For a separable complete locally convex space $E$, any sequentially continuous linear map $f:E’\to E’$ is continuous (where $E’$ is equipped with the weak*-topology).