A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions

In this paper we give general conditions on a countable family $V$ of weights on an unbounded open set $U$ in a complex Banach space $X$ such that the weighted space $HV\left(U\right)$ of holomorphic functions on $U$ has a Fréchet algebra structure. For such weights it is shown that the spectrum of $HV\left(U\right)$ has a natural analytic manifold structure when $X$ is a symmetrically regular Banach space, and in particular when $X=C^n$.