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Transactions of the American Mathematical Society

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On Kōmura's closed-graph theorem

Author: Michael H. Powell
Journal: Trans. Amer. Math. Soc. 211 (1975), 391-426
MSC: Primary 46A30
MathSciNet review: 0380339
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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.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1975 American Mathematical Society