A note on cotype of smooth spaces

Abstract

Abstract. For sufficiently smooth Banach spaces weak cotype 2 implies Hilbert space.It is well known that differentiability of the norm on a given Banach space X has a significant impact on the structure of X. Speaking about higher order differentiability of norms, it was shown in [2] that a Banach space X is isomorphic to a Hilbert space if on both X and X* there exist real valued functions ~ with bounded nonempty support and such that ~o' are locally Lipschitzian. It also is known that a Banach space X which admits a real valued twice continuously differentiable function ~o with bounded nonempty support is isomorphic to a FIilbert space provided any infinite dimensional subspace of X contains an infinite dimensional subspace which is isomorphic to a Hilbert space [9]. There are many spaces which do not admit real valued twice differentiable functions with bounded support and yet have norms with differential which is Lipschitzian on the sphere [12], I4],