  Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow
the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open
covers $\frak U$ of $M$ with a locally convex topology. Under certain
assumptions we prove that the canonical map $H^q(\frak U,\Cal S)\to H^q(M,\Cal
S)$ is an isomorphism of topological vector spaces.
