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Let (a) be a property of separated locally convex spaces. Call a locally convex space E [il] an (a )-space if y is the final topology defined by Iui: E [Tf] -i E} where each E.i[3] is an (a)-space. Then, for each locally convex space E[IJ], there is a weakest (a)-topology on E stronger that 5f, denoted fCa. Komura's closed-graph theorem states that the following statements about a locally convex space E [R1 are equivalent: (1) For every (a)-space F and every closed linear map u: F -4 EW[3, U is continuous. (2) For every separated locally convex topology %0 on E, weaker than X, we have J[ C a Much of this paper is devoted to amplifying Komura's theorem in special cases, some well-known, others not. An entire class of special cases, generalizing Adasch's theory of infra(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of "completeness" in the dual space.

On Kōmura’s closed-graph theorem