Grothendieck'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 $B_1, B_2\in S$, then there exists a $B_3\in S$ such that $B_3\supset B_1$ and $B_3\supset B_2$. Then $E_S$ is complete Iff algebraic linear functional $f(y)$ of $F$ that is weakly continuous on every $B\in S$ is expressed as $f(y)=\left\langle{x,y}\right\rangle{}$ for some $x\in E$. When $E_S$ is not complete, the space of all linear functionals satisfying this condition gives the completion $\hat E_S$ of $E_S$.

See also Mackey's Theorem

References

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