Differentiable functions on 𝑐₀

Abstract

If E and F are two Banach spaces, denote by C p ' q (E % F), O^g ^pti* 00 , those functions in C P (E, F) whose derivatives of order less than or equal to q are bounded. Call a Banach space, E, C PtQ smooth if there exists a nonzero C p,q function on E with bounded support. Then finite dimensional spaces are C 00 -00 smooth and if an L p space is C q smooth it is also C q ' q smooth. Although c 0 is known to possess a C (away from zero) norm as described in Bonic and Frampton [i ], it is a consequence of the following theorem that c Q is not C 2,2 smooth. THEOREM. Let fzC l (cQ, R) with Df uniformly continuous. Then the support off is unbounded.