Blid maps, a replacement for smooth bump functions, extend local smooth maps to global maps on spaces like C[0,1] and support a weakened bump-function condition in linearization theorems.
Belitskii, The Sternberg theorem for a Banach space, Funct
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New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples
Blid maps, a replacement for smooth bump functions, extend local smooth maps to global maps on spaces like C[0,1] and support a weakened bump-function condition in linearization theorems.