Local automorphisms are differential operators on some Banach spaces

Wells, John C.; DePrima, Charles R.

doi:10.1090/S0002-9939-1973-0324470-6

Local automorphisms are differential operators on some Banach spaces
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by John C. Wells and Charles R. DePrima PDF

Proc. Amer. Math. Soc. 40 (1973), 453-457 Request permission

Abstract:

If $E$ belongs to a certain category of Banach spaces (the ${B^\infty }$-smooth spaces) which include Hilbert spaces and if $F$ is any normed space, it is proved that any local linear automorphism of ${C^\infty }(E,F)$ is a differential operator. This generalizes a result of J. Peetre when $E = {R^n}$ and $F = R$.

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