Abstract
Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^infty-smooth bump which depends locally on finitely many coordinates.
Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^infty-smooth bump which depends locally on finitely many coordinates.