The main result in this paper is the following linearization theorem. For each open set U in a complex Banach space E, there is a complex Banach space G infinity(U) and a bounded holomorphic mapping g(U): U --> G infinity(U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping f: U --> F, there is a unique continuous linear operator T(f): G infinity(U) --> F such that T(f) omicron g(U) = f. The correspondence f --> T(f) is an isometric isomorphism between the space H infinity(U ; F) of all bounded holomorphic mappings from U into F, and the space L(G infinity(U); F) of all continuous linear operators from G infinity(U) into F. These properties characterize G infinity(U) uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces H infinity(U; F) and L(G infinity(U); F).3242867887