LetEbe a separable Banach spaceUbe a non-empty open subset ofEτ be a strictly lower semicontinuous function onUsuch thatd(x, ∂U) ≥ τ(x) forx∈U. LetBτ (U) denote the collection of all finite unions of closed balls Ƀp(x) with centerx∈Uand ρ < τ (x); and, letHτ(U) be the algeba of the complex holomorphic functions inUwhich are hounded in every set ofBτ(U) furnished with the Fréchet topology of the uniform convergence over the elements ofBτ(U).