Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively. A dual of $E^{\star}$ given by the $\beta(E^{\star\star}, E^{\star})$ topology is usually denoted by $E^{\star\star}$ and it is called a double dual of $E$.
Question 1
Is it true that the topology $\beta(E^{\star}, E^{\star\star})$ on $E^{\star}$ is always finer than $\beta(E^{\star}, E)$ (on $E^{\star}$), apart from the case when $E$ is reflexive and these topologies coincide.
Fact 1
It is well-known that if $E$ is an (F)-space then the (initial) (F)-space topology coincides with strong topology $\beta(E, E^{\star})$.
Question 2
Can we find a non-metrizable locally convex space for which the above sentence is true? (i.e. instead of (F)-space we put a non-metrizable locally convex space).
Question 3
Let $E$, $F$ be LCS spaces and let $A\colon E \rightarrow F$ be a linear mapping which is weakly continuous. Is it true that if $A^{\star \star}$ is continuous (in $\beta(E^{\star \star}, E^{\star \star})$ topology) then $A$ is continuous.
Thank you in advance for any help.
