The Envelope of Holomorphy of Riemann Domains over a Countable Product of Complex Planes

This paper deals with the problem of constructing envelopes of holomorphy for Riemann domains over a locally convex space. When this locally convex space is a countable product of complex planes the existence of the envelope of holomorphy is proved and the domains of holomorphy are characterized. For the Riemann domains over the cartesian product CN of a countable number of complex planes, the domains of holomorphy are characterized and the existence of the envelope of holomorphy is proved. Also, for Riemann domains over a complex separated locally convex space F such that the closed convex hull of every compact subset is compact, the existence of the normal envelope of holomorphy is proved. Let (U, q>) be a Riemann domain over E. This means that U is a connected separated topological space and <p is a local homeomorphism from U into E. If a £ U and A^E, a + A is defined by a + A = [cp\Wy\cp(a) + A), W being an open subset of U where cp is a homeomorphism, aeW, and cp(a) + A <^<p(W). When A has only one element h, a + h denotes the unique element of a + {//}. If B is a subset of U, B + A = \J(b + A) beB where b + A has the meaning just stated. A complex mapping / defined in U is holomorphic if, for every u in U, there is an open convex balanced neighborhood U of zero in F and a sequence of continuous «-homogeneous polynomials in E: (n!)"1 dnf(u), n = 0, I,..., such that «+t/<= £/and co f(u + h)= 2 (nl)-1 d»fC)(h), n = 0 the series converging uniformly for h in U. In the algebra Jf (U) of all complex Received by the editors July 6, 1971. AMS 1970 subject classifications. Primary 46A99; Secondary 58B10.