Is the strong topology of a locally convex space always 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?

 A: The answer to question 1 is negative. There are Frechet spaces $X$ whose strong duals $(X',\beta(X',X))$ are not barrelled (equivalently, not bornological by a theorem of Grothendieck). The first examples of such non-distinguished Frechet spaces were constructed by Köthe and Grothendieck but there are also examples which are very easy to describe: According to Taskinen the Frechet space $C(\mathbb R) \cap L^1(\mathbb R)$ of continuous Lebesgue-intergrable functions (endowed with the seminorms $\int|f(x)|dx + \sup\{|f(x)|: x\in [-n,n]\}$) is not distinguished. This answers question 1 with $E=(X',\sigma(X',X))$.
EDIT: If $M$ is a barrel in $E$ then its polar $M^\circ$ is a $\sigma(X,X')$-bounded subset of $X$ and hence it is bounded in the Frechet topology so that $M=M^{\circ\circ}$ is a $\beta(X',X)$-neighbourhood of $0$. Conversely, the typical $0$-neighbourhood $B^\circ$ in $\beta(X',X)$ with a bounded subset $B$ of $X$ is a barrel in $(X',\sigma(X',X))$.
A: Many examples of spaces $C_p(X)$ whose strong dual is not barrelled can be found in a recent paper "Examples of Nondistinguished Function Spaces $C_p(X)$", Journal of Convex Analysis, 2019.
A: No. Let $E$ be the space $C[0,1]$ of continuous functions on $[0,1]$ with the norm $\|x\|=\int_0^1 |x(t)|dt$. Then $E=E_\beta$ is normable and separable, but it is not barrelled. The set $B=\{x\in E: \sup_{t\in [0,1]} |x(t)|\le1\}$ is a barrell in $E$, but it is not neighborhood at zero.
