Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Open mappings and the lack of full completeness of $\mathcal {D}’(\Omega )$
HTML articles powered by AMS MathViewer

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

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
  • N. Bourbaki, Eléments de mathématique. XIX. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. (Fascicule de résultats.), Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1230, Hermann & Cie, Paris, 1955 (French). MR 0077883
  • Charles Harvey, On domination estimates and global existence, J. Math. Mech. 16 (1967), 675–702. MR 0214903
  • Lars Hörmander, On the range of convolution operators, Ann. of Math. (2) 76 (1962), 148–170. MR 141984, DOI 10.2307/1970269
  • —, Linear partial differential operators, Die Grundlehren der math. Wissenshaften, Band 116, Academic Press, New York and Springer-Verlag, Berlin and New York, 1963. MR 28 #4221.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46.01, 35.00
  • Retrieve articles in all journals with MSC: 46.01, 35.00
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