COMPLETE SPACES OF VECTOR-VALUED HOLOMORPHIC GERMS

Let K be a non-empty compact subset of a Frechet space E and let X be a Banach space. By means of a given representation of the LB-space H(K, X) of germs of holomorphic functions with values in X as a space of linear operators, it is proved that the space H(K, X) is complete if E is quasinormable or if X is complemented in its bidual. If E is a Frechet-Montel space, X is an L infinity-space in the sense of Lindenstrauss and Pelczyhski and H(K,X) is complete, then E'(b) ($) over cap circle times(epsilon)X must be an LB-space. It is an open problem whether c(0)(E'(b)) similar or equal to E'(b) ($) over cap circle times(epsilon)c(0) is an LB-space for every Frechet-Montel space E.