Non-quasianalytic curves in Fréchet spaces
The author studies non-quasianalytic curves in locally convex spaces, i.e., vector valued \(\omega\)-ultradifferentiable functions \(f:\Omega\to E\) in the setting given by [\textit{R. W. Braun, R. Meise} and \textit{B. A. Taylor}, ``Ultradifferentiable functions and Fourier analysis'', Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)], where \(\Omega\) is an open subset of \(\mathbb{R}^n\) and \(E\) is a locally convex space. In 1990, \textit{A. Kriegl} and \textit{P. W. Michor} [``The convenient setting for real analytic mappings'', Acta Math. 165, No. 1--2, 105--159 (1990; Zbl 0738.46024)] introduced several notions of real analytic curves in a locally convex space \(E\). Since 1998, Bonet and Domański have investigated in different papers real analytic curves with values in Fréchet and (LB)-spaces, clarifying the relation between various notions of real analyticity introduced by Kriegl and Michor (see, for example, [\textit{J. Bonet} and \textit{P. Domański}, ``Real analytic curves in Fréchet spaces and their duals'', Monatsh. Math. 126, No. 1, 13--36 (1998; Zbl 0918.46034)]), and have applied their results to investigate surjective linear partial differential operators between spaces of vector valued real analytic functions and parameter dependence of the solutions of these operators. The author solves a problem posed by Kriegl and Michor in a meeting celebrated in Paderborn (Germany) in November 2008 with the characterization of different notions of non-quasianalytic curves which take values in a Fréchet or (LB)-space.
vector valued ultradifferentiable functions
scientific article; zbMATH DE number 5988483
scientific article