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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?

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    $\begingroup$ The space of tempered distributions with its strong topology is barrelled, but not with the weak* topology: en.wikipedia.org/wiki/Barrelled_space (see Properties) $\endgroup$
    – Dan
    Commented Oct 2, 2012 at 15:44

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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.

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