\((LB^{\infty})\)-structure of spaces of germs of holomorphic functions
For a nuclear Fréchet space \(E\) and a balanced convex compact subset \(B\) of \(E\) denote by \(H(B)\) the space of germs of holomorphic functions on \(B\). Then \(H(B)'\) is a nuclear Fréchet space. Consider the following assertions: (1) \(E\) has the property \((\widetilde\Omega_B)\). (2) \(H(B)'\) has the property \((LB^\infty)\). (3) The set \(B\) is not pluripolar in \(E\). The author proves that (1) implies (2) and that (1)--(3) are equivalent, provided that \(E\) has the bounded approximation property. Under the latter hypothesis the equivalence of (1) and (3) was shown in \textit{S. Dineen}, \textit{R. Meise} and \textit{D. Vogt} [Bull. Soc. Math. Fr. 112, 41-68 (1984; Zbl 0556.46003)]. Ideas from this paper and from \textit{D. Vogt} [J. Reine Angew. Math. 345, 182-200 (1983; Zbl 0261.46003)] are used in the proof. The author also shows that if \(E\) has a Schauder basis, the space \(H(B)'\) satisfies \((\widetilde\Omega_B)\) (resp. \((\overline\Omega_B)\)) if and only if \(E\) has the same property.
linear topological invariants of \(\Omega\)-type
scientific article; zbMATH DE number 1505970