Locally flat Banach spaces

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates. This work was supported by the research project MSM 0021620839 and by the grant of the Grant Agency of the Czech Republic No. 201/05/P582 and No. 201/06/0018.