## Grothendieck's Theorem

Let and be paired spaces with a family of absolutely convex bounded sets of such that the sets of generate and, if , then there exists a such that and . Then is complete Iff algebraic linear functional of that is weakly continuous on every is expressed as for some . When is not complete, the space of all linear functionals satisfying this condition gives the completion of .

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