  This paper considers two commuting smooth transformations on a Banach space,
and proves the sub-additivity of the measure theoretic entropies under mild
conditions. Furthermore, some additional conditions are given for the equality
of the entropies. This extends Hu's work about commuting diffeomorphisms in a
finite dimensional space (Huyi Hu, 1993, Ergod. Th. Dynam. Sys., \textbf{13}:
73-100) to the case of systems on an infinite dimensional Banach space.
