The concept can be generalised from cartesian spaces to Banach
spaces and some other infinite-dimensional spaces. There is a
locale-based analogue suitable for constructive mathematics
which is not described as a function of points but as a special
case of a continuous map (in the localic sense). A manifold
whose transition functions are smooth functions is a smooth
manifold. The category Diff is the category whose objects are
smooth manifolds and whose morphisms are smooth functions
betweeen them. Yet more generally, the morphisms between
generalised smooth spaces are smooth maps. For functions between
manifolds that fall short of full smoothness, see differentiable
map. Basic facts about smooth functions are the Hadamard lemma A
crucial property of smooth functions, however, is that they
contain also bump functions. Examples of sequences of
infinitesimal and local structures |first order
infinitesimal||formal = arbitrary order infinitesimal||local =
stalkwise||finite| |derivative||Taylor series||germ||smoot