On Kōmura’s closed-graph theorem

Abstract:

Let $(\alpha )$ be a property of separated locally convex spaces. Call a locally convex space $E[\mathcal {J}]$ an $(\bar \alpha )$-space if $\mathcal {J}$ is the final topology defined by ${\{ {u_i}:{E_i}[{\mathcal {J}_i}] \to E\} _{i \in I}}$, where each ${E_i}[{\mathcal {J}_i}]$ is an $(\alpha )$-space. Then, for each locally convex space $E[\mathcal {J}]$, there is a weakest $(\bar \alpha )$-topology on E stronger that $\mathcal {J}$, denoted ${\mathcal {J}^{\bar \alpha }}$. Kōmura’s closed-graph theorem states that the following statements about a locally convex space $E[\mathcal {J}]$ are equivalent: (1) For every $(\alpha )$-space F and every closed linear map $u: F \to E[\mathcal {J}]$, u is continuous. (2) For every separated locally convex topology ${\mathcal {J}_0}$ on E, weaker than $\mathcal {J}$, we have $\mathcal {J} \subset \mathcal {J}_0^{\bar \alpha }$. Much of this paper is devoted to amplifying Kōmura’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.