# Strong topology

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).

Thank you in advance for any help.

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For the first question, the strong topology is the polar topology generated by all weakly bounded subsets. The weakly bounded subsets of $E$ are also weakly bounded in $E^{\ast\ast}$ since $E\subset E^{\ast\ast}$ and they have the same dual space $E^{\ast}.$ Therefore $\beta(E^{\ast},E^{\ast\ast})$ is finer than $\beta(E^{\ast},E).$
I have found a theorem in "Topological vector spaces" of Adash, Ernst and Keim, that a lcs space $(E, \tau)$ is barelled if and only if $\tau=\beta(E, E^*)$. This answers the Question 2 completely in the class of lcs spaces. – Tomek Kania Aug 17 '11 at 0:29