Quasianalytic functionals and projective descriptions. (English) Zbl 1064.46018

Bonet, José; Meise, Reinhold

Math. Scand. 94, No. 2, 249-266 (2004).

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.

Reviewer: A. Albanese (Lecce)

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

Keywords:

projective description; ultradifferentiable space of Roumieu type; quasianalytic weight function

Citations:

Zbl 0978.46020