On compact and bounding holomorphic mappings

Let E and F be complex Banach spaces. We say that a holomorphic mapping f from E into F is compact respectively bounding if f maps some neighbourhood of every point of E into a relatively compact respectively bounding subset of F. Recall that a subset of E is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on E. Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.