The Dolbeault complex in infinite dimensions. II
  We prove that the equation d-bar u = f can be solved on a ball B(R) of radius R in the Banach space l^1 if f is a closed Lipschitz continuous (0,1) form on B(R).   We also present examples of closed (0,1) forms f of various regularities on the spaces l^p that are not exact. In particular, in the first result above, it is not enough to assume that f is merely continuous, rather than Lipschitz continuous. 
Laszlo Lempert