Geometry of Lebesgue-Bochner function spaces—smoothness

Abstract

There exist real Banach spaces E such that the norm in E is of class C' away from zero; however, for any p, I < p < oo, the norm in the Lebesgue-Bochner function space LP(E, ,u) is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces. Introduction. The class of Lebesgue-Bochner function spaces, introduced by Bochner and Taylor [4] in 1938, has been found to be of considerable importance in various branches of mathematics, and is discussed at length in Dinculeanu [11], Dunford and Schwartz [12], and Edwards [13]. The study of the geometric properties of the Lebesgue-Bochner function spaces dates back about three decades: Day [8] and McShane [17], respectively, characterized uniform convexity and smoothness of these spaces. In fact, the only known result concerning the smoothness of the Lebesgue-Bochner function spaces is due to McShane, and his result concerns only the directional derivative (Gateaux derivative) of the norm in this class of Banach spaces. Even the Frechet differentiability of the norm has not been considered anywhere. It might be mentioned in this connection that the first systematic study of higher-order differentiability of the norm in a Banach space was made by Kurzweil [15] in 1954. Subsequently, in 1965, Bonic and Frampton [5a] extended Kurzweil's results, and in 1966 [5b] they discussed various categories of smooth Banach manifolds. In 1967, Sundaresan [18] extended some of Kurzweil's results independently. In [5b] and [18], the order of differentiability of the norm in the classical Lp spaces, 1 < p < oo, is obtained; while in Sundaresan [20], the smoothness of the norm in C(X, E) is discussed. For an elegant up-to-date account of smooth Banach spaces, and related concepts one might refer to the lecture notes by S. Yamamuro [23]. This paper contains the first systematic investigation of the higher-order differentiability of the norm function in the LebesgueBochner function spaces Received by the editors April 28, 1973. AMS (MOS) subject classifications (1970). Primary 46E40, 28A45; Secondary 58C20, 28A15.