  The question of extension of locally defined maps to the entire space arises
in many problems of analysis (e.g., local linearization of functional
equations). A known classical method of extension of smooth local maps on
Banach spaces uses smooth bump functions. However, such functions are absent in
the majority of infinite-dimensional spaces. We suggest a new approach to
localization of Banach spaces with the help of locally identical maps, which we
call blid maps. In addition to smooth spaces, blid maps also allow to extend
local maps on non-smooth spaces (e.g., $C^q [0, 1]$, $q=0, 1, 2,...$).
  For the spaces possessing blid maps, we show how to reconstruct a map from
its derivatives at a point (see the Borel Lemma). We also demonstrate how blid
maps assist in finding global solutions of cohomological equations having
linear transformation of the argument. We present application of blid maps to
local differentiable linearization of maps on Banach spaces.
  We discuss differentiable localization for metric spaces (e.g.,
$C^{\infty}(\R)$), prove an extension result for locally defined maps and
present examples of such extensions for the specific metric spaces. In
conclusion, we formulate open problems.
