Markuševič bases in some dual spaces

Abstract

If X* is locally uniformly rotund, then X* has a Markusevic basis. It was proved by Tacon [8] that if the norm of a Banach space X is Frechet differentiable, then X* admits a projectional resolution {Pa} of the identity I. The same result was shown in [43 if X admits a continuously Fre'chet differentiable function with bounded nonempty support. Here we prove that if X* is locally uniformly rotund, then the projections (P a+1a) have also nice ranges for transfinite induction, which enables us to derive the result mentioned in the Abstract. The above mentioned results lead to the following questions: (a) Does X* have a Markusevic basis if X has Fre'chet differentiable norm? (b) Does there exist a locally uniformly rotund Banach space without Markusevic basis? (There is a strictly convex space without Markusevic basis by [21.) Also, it should be remarked that the result mentioned in the Abstract cannot be strengthened to give a Markusevic basis on X* with coefficients from X (i.e. shrinking Markusevic basis on X), since there is a Banach space X which is not weakly compactly generated and X* is locally uniformly rotund [6]. A Banach space X is locally uniformly rotund (LUR) if whenever xn$ x EX, ?xnI = |x| = 1, lim Ixn+ xi = 2, then lim Ixn x| = O. If M is a subset of a Banach space X, then cl M means the norm closure of M in X and sp M is the norm closed linear hull of M. dens X is the smallest cardinality of a dense subset of X. Symbol _ denotes the linear isometry. A Markusevic' basis in a Banach space X is a family lxal C X, {lal C X a E eF such that /a(Xi8) = 3a,3, sp Uaxa = X and 1/al total on X (i.e. fla /a(0) = 0). Unless stated otherwise, in X* we take the dual sup-norm. Theorem. If X* is LUR, then there is a transfinite sequence of continuous linear projections Qa: X* -x* XO < a < v with (1) Qo = 0, Qv = I; (2) QaQ3 = Q8Qa = QQa if ca < 1; Received by the editors April 24, 1974. AMS (MOS) subject classifications (1970). Primary 46B99.