Mackey's Theorem

Let $E$ and $F$ be paired spaces with $S$ a family of absolutely convex bounded sets of $F$ such that the sets of $S$ generate $F$ and, if $B_1, B_2\in S$, there exists a $B_3\in S$ such that $B_3\supset B_1$ and $B_3\supset B_2$. Then the dual space of $E_S$ is equal to the union of the weak completions of $\lambda B$, where $\lambda>0$ and $B\in S$.

See also Grothendieck's Theorem

References

Iyanaga, S. and Kawada, Y. (Eds.). ``Mackey's Theorem.'' §407M in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1274, 1980.