ABSTRACT
Let U be an open subset of a complex locally convex space E, and let ℋ(U) denote the algebra of all holomorphic functions on U, with the compact-open topology.
If U is an open subset of C n , then it is well known that the following conditions are equivalent:
(a) U is a domain of holomorphy.
(b) Every continuous homomorphism T : ℋ(U) → C is an evaluation.
(c) Every finitely generated, proper ideal of ℋ(U) has a common zero.
This is a survey of what is known in this direction in the case of infinite dimensional spaces.
