Smooth Bump Functions and Renormings of Banach Spaces: A Survey

Throughout this survey X will denote a real Banach space, X^* its continuous dual, B(X) and S(X) its closed unit ball and sphere, respectively. If X is finite dimensional, it is well known that there exists a function f: X →ℝ (the reals) which is of class C^k (in fact C^∞) and f has bounded nonempty support, that is, cℓ{x ∈ X: f(x) ≠ 0} is nonempty and bounded. We call such a function a C^k -bump function on X. On the other hand, if X is infinite dimensional, the situation is more delicate.