Peetre characterized local operators defined on the smooth section space over an open subset of an Euclidean space as ``linear differential operators'. We look for an extension to such maps of smooth vector sections of smooth Banach bundles. Since local operators are special disjointness preserving operators, it leads to the study of the disjointness structure of smooth Banach manifolds. In this thesis, we take an abstract approach to define the``smooth functions', via the so-called S-category. Especially, it covers the standard classes C^{n} and local Lipschitz functions, where 0≤n≤¡Û. We will study the structure of disjointness preserving linear maps between S-smooth functions defined on separable Banach manifolds. In particular, we will give an extension of Peetre's theorem to characterize disjointness preserving linear mappings between C^n or local Lipschitz functions defined on locally compact metric spaces.