I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak\* topology. Is is a Barreled space, say, a space whose convex, balanced, absorbing and closed subsets are neighborhood of the origin? 



A question on Schwartz distributions
If the weak$^\*$ dual of a locally convex space $X$ is barrelled then the bounded sets of $X$ are finite dimensional (because the polar of a bounded set $B$ is a barrel which then contains the polar of a finite set $E$ and
the theorem of bipolars implies that $B$ is contained in the absolutely convex hull of $E$). For Frechet spaces (like the Schwartz space) this implies that $X$ is finite dimensional.


