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Open mappings and the lack of full completeness of $ \mathcal{D}'(\Omega)$

Authors: Charles Harvey and F. Reese Harvey
Journal: Proc. Amer. Math. Soc. 25 (1970), 786-790
MSC: Primary 46.01; Secondary 35.00
MathSciNet review: 0265906
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Abstract: Consider a linear map $ T$ of one locally convex linear space into another which is densely defined and has a closed graph. We characterise the property that $ T$ is an open map in terms of two properties of its adjoint map $ {T^{\ast}}$. These results are used to show that if $ \Omega $ is an open subset of $ {R_n}$ for which there is a linear constant coefficient differential operator $ P$ such that $ \Omega $ is $ P$-convex but not strongly $ P$-convex, then (i) $ \mathcal{D}'(\Omega )$ is not fully complete, (ii) the range of the adjoint map $ ^tP$ is closed but not bornological.

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Keywords: Locally convex linear space, open linear map, equicontinuous subset, fully complete space, Mackey space, $ \mathcal{D}'(\Omega )$, $ P$-convex, strongly $ P$-convex, bornological
Article copyright: © Copyright 1970 American Mathematical Society

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