For a locally convex space $E$ let $E_\beta$ be the space $E$ endowed with the locally convex topology $\beta(E,E')$ whose neighborhood base at zero consists of barrels, i.e., closed absolutely convex absorbing sets.

Observe that a space $E$ is barrelled if and only if $E=E_\beta$.

Question 1. Is $(E_\beta)_\beta=E_\beta$ for any locally convex space? Equivalently, is the space $E_\beta$ always barrelled?

If not, then we can ask a more restricted version of Question 1.

Question 2. Let $E$ be a locally convex space such that the space $E_\beta$ is normable. Is $E_\beta$ barrelled?

For a locally convex space $E$ let $E_\beta$ be the space $E$ endowed with the locally convex topology $\beta(E,E')$ whose neighborhood base at zero consists of barrels, i.e., closed absolutely convex absorbing sets.

Observe that a space $E$ is barrelled if and only if $E=E_\beta$.

Question 1. Is $(E_\beta)_\beta=E_\beta$ for any locally convex space? Equivalently, is the space $E_\beta$ always barrelled?

If not, then we can ask a more restricted version of Question 1.

Question 2. Let $E$ be a locally convex space such that the space $E_\beta$ is normable (and separable). Is $E_\beta$ barrelled?