  A known classical method of extension of smooth local maps of Banach spaces
uses smooth bump functions. However, such functions are absent in the majority
of infinite-dimensional Banach spaces. This is an obstacle in the development
of local analysis, in particular in the questions of extending local maps onto
the whole space. We suggest an approach that substitutes bump functions with
special maps, which we call blid maps. It allows us to extend smooth local maps
from non-smooth spaces, such as $C^q[0,1], q=0,1,...$. As an example of
applications, we show how to reconstruct a map from its derivatives at a point,
for spaces possessing blid maps. We also show how blid maps can assist in
finding global solutions to cohomological equations having linear
transformation of argument.

Functional Analysis