COLLOQU IUM MATHEMAT ICUM VOL. LXIV 1993 FASC. 1 WEAK MEROMORPHIC EXTENSION BY

The relation between weak extensibility and extensibility of vector-valued holomorphic functions on open sets and on compact sets has been investi-gated by many authors, for example Ligocka and Siciak [6] for open sets in a metric vector space, Siciak [9] and Waelbroeck [10] for compact sets in Cn, N. V. Khue and B. D. Tac [8] for compact sets in a nuclear metric vector space. The aim of the present note is to prove some results for Banach-valued meromorphic functions on open sets and on compact sets in Cn. We recall [7] that a holomorphic function f on a dense open subset G0 of an open set G in Cn with values in a sequentially complete locally convex space F is called meromorphic on G if for each z ∈ G there exists a neighbourhood U of z and holomorphic functions g and σ on U with values in F and C respectively such that f |G0 ∩ U = g/σ|G0 ∩ U. Put P (f) = {z ∈ G: f is not holomorphic at z}. It is known [7] that P (f) is either empty or a hypersurface in G. Finally, for each open subset G of Cn, we denote by G ̂ the envelope of holomorphy of G. First we prove the following Theorem 1. Let G be an open set in Cn and F a Banach space. Assume that f is an F-valued meromorphic function on an open subset X of G such that x∗f can be extended to a meromorphic function x̂∗f on G for all x ∗ ∈ F ∗, the dual space of F. Then f can be meromorphically extended to G. P r o o f. It suffices to show that f can be meromorphically extended through every point z ∈ ∂X. Fix z0 ∈ ∂X. Put B = a1,..., as; b1,..., bt ∈ (Q+ iQ)n