The weak nullstellensatz for finite dimensional complex spaces

Two of the most important global properties of complex spaces (X, O), holomorphic convexity and holomorphic separability, can each be characterized in terms of the standard natural map X: X→ S_c(O(X)), x→X_x X_x(f):=f(x), f∈ O(X) of X into the continuous spectrum S_c(O(X)) of the global function algebra O(X) The question as to whether there is any global function theoretical property of (X, O) corresponding to the surjectivity of X has remained unanswered.The purpose of this paper is to present an answer for finite dimensional spaces. For such spaces (X, O) it will be shown that the surjectivity of X is equivalent to requiring that for finitely many functions fl9 -9fme^(X) with no common zero on X there exist functions f₁…, f_m ∈ O(X) with this property will be called the weak Nullstellensatz for the complex space (X, O). An example due to H. Rossi shows that this result is not valid for infinite dimensional complex spaces. An application of the weak Nullstellensatz for Fréchet algebras A involving the Michael conjecture is that S_c(A) is always dense in the spectrum S(A) of A.