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Proceedings of the American Mathematical Society

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Linear operators commuting with translations on $ \mathcal{D}(\mathbf{R})$ are continuous


Author: Gary Hosler Meisters
Journal: Proc. Amer. Math. Soc. 106 (1989), 1079-1083
MSC: Primary 47B38; Secondary 46F10
DOI: https://doi.org/10.1090/S0002-9939-1989-0979227-1
MathSciNet review: 979227
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Abstract: Let $ \mathcal{D}\left( {\mathbf{R}} \right)$ denote the Schwartz space of all $ {C^\infty }$-functions $ f:{\mathbf{R}} \to {\mathbf{C}}$ with compact supports in the real line $ {\mathbf{R}}$. An earlier result of the author on the automatic continuity of translation-invariant linear functionals on $ \mathcal{D}\left( {\mathbf{R}} \right)$ is combined with a general version of the Closed-Graph Theorem due to A. P. Robertson and W. J. Robertson in order to prove that every linear mapping $ S$ of $ \mathcal{D}\left( {\mathbf{R}} \right)$ into itself, which commutes with translations, is automatically continuous.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0979227-1
Keywords: $ \mathcal{D}\left( {\mathbf{R}} \right)$, $ {C^\infty }$-functions of compact support, automatic continuity, translation-invariant linear functionals, operators commuting with translations, translation-invariance, Closed-Graph Theorem, Fourier transform, convolution, distributions of compact support, inductive limit of Fréchet spaces
Article copyright: © Copyright 1989 American Mathematical Society