The Dolbeault Complex in Infinite Dimensions Ii

The goal of this series is to explore various aspects of the inhomogeneous Cauchy– Riemann, or ∂, equation on infinite dimensional complex manifolds. In the first paper in the series we argued for the importance of such an undertaking; we also gave rather complete results when the manifold in question is an infinite dimensional projective space; see [L]. In the present work we turn to the analytically more challenging problem of solving the ∂ equation in an open subset of a Banach space, and offer a positive result in the space l. Up to now not a single infinite dimensional Banach space and an open set therein have been proposed where the equation