Peetre's theorem in the locally convex setting
\textit{J. Peetre}'s theorem [Math. Scand. 7, 211--218 (1959; Zbl 0089.32502)] characterizes linear partial differential operators as local linear maps \(T\) between spaces of \(C^\infty\) functions without mentioning any sort of derivatives. It became a relevant tool in several fields of mathematics. In the paper under review, the author generalizes this theorem in several ways for operators between spaces of \(C^r\) functions defined on a finite-dimensional domain with values in infinite-dimensional locally convex spaces. For general range spaces, the continuity of the operator \(T\) must be assumed. On the other hand, it is proved that local linear operators are necessarily continuous if the smooth functions of class \(r\) take values in Fréchet spaces. A variant of Peetre's theorem for continuous functions defined on a metrizable space without isolated points is also presented.
infinite-dimensional differential calculus
scientific article; zbMATH DE number 6479273