Open mappings and the lack of full completeness of $\mathcal {D}’(\Omega )$
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- by Charles Harvey and F. Reese Harvey
- Proc. Amer. Math. Soc. 25 (1970), 786-790
- DOI: https://doi.org/10.1090/S0002-9939-1970-0265906-6
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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.References
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Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 25 (1970), 786-790
- MSC: Primary 46.01; Secondary 35.00
- DOI: https://doi.org/10.1090/S0002-9939-1970-0265906-6
- MathSciNet review: 0265906