Linear operators commuting with translations on 𝒟(𝐑) are continuous

Abstract

Let D ( R ) \mathcal {D}\left ( {\mathbf {R}} \right ) denote the Schwartz space of all C ∞ {C^\infty } -functions f : R → C f:{\mathbf {R}} \to {\mathbf {C}} with compact supports in the real line R {\mathbf {R}} . An earlier result of the author on the automatic continuity of translation-invariant linear functionals on D ( R ) \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 S of D ( R ) \mathcal {D}\left ( {\mathbf {R}} \right ) into itself, which commutes with translations, is automatically continuous.