Analytic Norms in Orlicz Spaces

It is shown that an Orlicz sequence space hM admits an equivalent analytic renorming if and only if it is either isomorphic to l2n or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent C∞-Fréchet norm, but no equivalent analytic norm. In this note, we denote by hM as usual the subspace of an Orlicz sequence space lM generated by the unit vector basis. More terminology and notation concerning Orlicz spaces can be found in [LT]. Let us also point out that by C-smoothness (or analyticity) of a norm we always mean away from the origin (as is usual in renorming theory). The characterization of the best order of C-Fréchet smoothness of some renorming, k ∈ N ∪ {+∞}, for hM was obtained in [M], [MT1], [MT2]. In our present note, we complete the characterization also for analytic renormings. We show that an Orlicz sequence space hM has an analytic renorming if and only if hM ∼= l2n, n ∈ N, or hM is isomorphically polyhedral. Let us recall that a separable Banach space X is isomorphically polyhedral if it has an equivalent polyhedral norm. By a theorem of Fonf [F], this is the case if and only if X admits an equivalent norm with a countable boundary. More precisely, there exists a sequence {fi}n∈N in X∗ such that ‖x‖ = max{|fi(x)|, i ∈ N}. According to one of the results from [DFH], we have the following: Theorem 1. Every separable isomorphically polyhedral Banach space X admits an equivalent analytic form. We prove that the converse is also true if we impose additional conditions on the space X . In connection with our result it should be noted that by recent work of Gonzalo and Jaramillo ([GJ]) every separable Banach space with a symmetric basis and C∞-Fréchet smooth norm is isomorphic to l2n, provided it does not contain a copy of c0. Our approach is entirely different from that in [MT1] and relies on methods from [DFH] and [H1]–[H3]. As a corollary, relying on an example of Leung [L], we show that there exists a separable Banach space with C∞-Fréchet smooth norm which Received by the editors March 30, 1998 and, in revised form, November 25, 1998. 2000 Mathematics Subject Classification. Primary 46B03, 46B45.