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Quasianalytic functionals and projective descriptions. (English) Zbl 1064.46018
Let \(\omega\) be a weight function and let \({\mathcal E}_{\{\omega\}}(G)\) denote the space of all \(\omega\)-ultradifferentiable functions of Roumieu type on a convex open set \(G\) in \(\mathbb R^N\). The strong dual \({\mathcal E}_{\{\omega\}}(G)'_b\) of \({\mathcal E}_{\{\omega\}}(G)\) is isomorphic to an (LF)-space \({\mathcal F E}'_{\{\omega\}}(G)\) of entire functions on \({\mathbb C}^N\) via the Fourier–Laplace transform. When \(\omega\) is a non-quasianalytic weight function, the authors showed in [J. Math. Anal. Appl. 255, 122–136 (2001; Zbl 0978.46020)] that the natural (LF)–topology on \({\mathcal F E}'_{\{\omega\}}(G)\) cannot be described by means of the associated weighted sup-seminorms. In this paper, they extend this result to the case that \(\omega\) is a quasianalytic weight function.

MSC:
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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