Strict convexity and smoothness of normed spaces

Abstract

1. Definitions and outline of results. This paper contains the first examples of normed spaces not isomorphic to strictly convex or smooth spaces. The table below shows the properties now known to be possessed by a number of special Banach spaces; some conclusions and unsolved problems are discussed after the table. ?2 contains the positive results which enable us to show that the special examples do have the properties asserted, while ?3 contains the specific calculations which show that some spaces can not be made smooth or strictly convex. ?4 contains an example related to the impossibility of some generalizations of theorems of Kakutani and Michael on simultaneous extension of continuous functions; the proof presented for this example shows that this topic is, slightly, related to the one discussed in detail here. The following definitions and notation are used throughout the paper. DEFINITIONS. Let B be a normed linear space. If every chord of the unit sphere of B has its midpoint below the surface of the unit sphere, then B is called strictly convex (written SC); if through every point of the surface of the unit sphere of B there passes a unique hyperplane of support (that is, a tangent hyperplane) of the unit sphere, then B is called smooth (written SM); if both occur, then B is called SCM. If B is isomorphic to an SX space, X = C, M, or CM, then B is called an sx space. If I is an index set, let K = K(I) be the Tyhonov cube; that is, the topological product PiErJ;, where each J is the closed interval [-1, 1]. For p> 1 let Hp =Hp(I) be the set of those elements x of K(I) such that