A closed subspace of $\mathcal {D}(\Omega )$ which is not an $\mathrm {LF}$-space
HTML articles powered by AMS MathViewer
- by M. J. Kascic and B. Roth
- Proc. Amer. Math. Soc. 24 (1970), 801-802
- DOI: https://doi.org/10.1090/S0002-9939-1970-0257689-0
- PDF | Request permission
Abstract:
With proper choice of region $\Omega \subseteq {R^n}$ and constant coefficient linear partial differential operator $P$, namely $\Omega$ being $P$-convex but not strong $P$-convex, the range of $P$ in $\mathcal {D}(\Omega )$ is a closed subspace of $\mathcal {D}(\Omega )$ whose subspace topology differs from its canonical ${\text {LF}}$-topology. In the present paper this result is proved and an example of a pair $\Omega ,\;P$ satisfying the above hypotheses is presented.References
- Jean Dieudonné and Laurent Schwartz, La dualité dans les espaces $\scr F$ et $(\scr L\scr F)$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61–101 (1950) (French). MR 38553
- Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542 L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Bd. 116, Academic Press, New York and Springer-Verlag, Berlin and New York, 1963. MR 28 #4221.
- Taqdir Husain, The open mapping and closed graph theorems in topological vector spaces, Clarendon Press, Oxford, 1965. MR 0178331, DOI 10.1007/978-3-322-96210-2
- François Trèves, Linear partial differential equations with constant coefficients: Existence, approximation and regularity of solutions, Mathematics and its Applications, Vol. 6, Gordon and Breach Science Publishers, New York-London-Paris, 1966. MR 0224958
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 24 (1970), 801-802
- MSC: Primary 46.01; Secondary 35.00
- DOI: https://doi.org/10.1090/S0002-9939-1970-0257689-0
- MathSciNet review: 0257689