The regularity of spaces of germs of \(f\)-valued holomorphic functions
It is known for a long time that the space \(H(K)\) of scalar-valued holomorphic germs on compact subsets \(K\) of a Banach space \(E\) is always regular. In the paper under review the regularity of spaces \(H(K,F)\) of holomorphic germs on compact subets of an locally convex space \(E\) with values in a Fréchet space \(F\) is investigated. It comes out that the properties \((DN)\) and \((LB_\infty)\) of \(F\) play an essential role. These properties have been introduced by \textit{D. Vogt} [Arch. Math. 45, 255-266 (1985; Zbl 0621.46001)]. As main results the following theorems are proven. Theorem A. If \(F\) is a reflexive Fréchet space with property \((LB_\infty)\) and \(E\) a quotient space of the power series space of infinite type, then \(H(K,F)\) is regular for any unique compact \(K\) in \(E\). Theorem B. A Fréchet space \(F\) has \((DN)\) if and only if \(H(K,F)\) is regular for all compact subsets \(K\) of \(\mathbb{C}^{\mathbb{N}}\). The proof of Theorem B is based on the following lemma which seems to be of interest for itself. Lemma. Let \(F\) be a Fréchet space with a continuous norm and suppose that \(H(0,F)\) for \(0\in \mathbb{C}^{\mathbb{N}}\) is regular. Then \(H(K,F)\) is regular for all compact subets \(K\) of \(\mathbb{C}^{\mathbb{N}}\).
scientific article; zbMATH DE number 1239972