Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that if X is a Banach space and if there is a non-zero real-valued C ^∞-smooth function on X with bounded support, then either X contains an isomorphic copy of c ₀(N), or there is an integer k greater than or equal to 1 such that X is of exact cotype 2 k and, in this case, X contains an isomorphic copy of l ^2k(N). We also show that if X is a Banach space such that there is on X a non-zero real-valued C ⁴-smooth function with bounded support and if X is of cotype q for q<4, then X is isomorphic to a Hilbert space.