Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), {\rm WOT})$ is a fairly nice locally convex space. I am interested in the weak topology of this space. 


1) Does $\sigma\big( (B(E), {\rm WOT} ), (B(E), {\rm WOT} )^\*)={\rm WOT}$?


2) Is $(B(E), {\rm WOT} )$ a barelled space?



Weak topology of WOT
As Gerald suggests, the answer to question 1) is yes. The answer to the second question is negative (if $E$ is infinite-dimensional) because the operator norm makes $B(E)$ a Banach space and, on any Banach space $X$, there is no strictly coarser locally convex Hausdorff topology $\tau$ which is barrelled.


Otherwise, the closure of the unit ball in the coarser topology would be a $\tau$-neighborhood of $0$, that is, the identity $(X,\|\cdot\|) \to (X,\tau)$ is
*almost open* and thus open by the open mapping theorem (in a version which misses in most textbooks although the standard proof of the OMT gives this result).


