Dear MOs,

Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-topology induced by $\mathcal{D}(R)$, which makes it into a locally convex space (See Rudin Functional Analysis P.160).

The question is: Does the double dual space $\mathcal{D}''(R)$, i.e., the dual space of the distributions $\mathcal{D}'(R)$, exist? For Banach space $X$, we know that double dual $X^{**}\supseteq X$. They are equal when $X$ is reflective. A first guess is that $\mathcal{D}''(R)\supseteq \mathcal{D}(R)$.

Thanks for any comments! :-)

Anand