Normal structure in BochnerLp-spaces

Abstract

The concept of normal structure, a geometric property of sets in normed linear spaces, was introduced in 1948 by M. S. Brodskiϊ and D. P. Mil'man in order to study the existence of common fixed points of certain sets of isometries. Since then, normal structure has been studied both as a purely geometric property of normed linear spaces and as a tool in fixed point theory [1, 4, 5, 6]. In 1968, L. P. Belluce, W. A. Kirk, and E. F. Steiner [1] proved that the l°°-direct sum of two normed linear spaces with normal structure has normal structure. They were not however able to decide if normal structure is preserved under an l-direct sum of two normed linear spaces for 1 < p < oo. In 1984, T. Landes [5] proved that, if 1 < p < oo, normal structure is preserved under finite or infinite l-direct sums. In this article, the corresponding theorem is proven in the nondiscrete setting; consequently it is shown that, if (Ω, Σ, μ) is any measure space and 1 < p < oo, the Bochner Z/-space L(μ,X) has normal structure exactly when X has normal structure. A normed linear space X has normal structure if, for each closed bounded convex set in X that contains more than one point, there is a point p in such that sup{||/? x | | : x e K} is less than the diameter of K such a point p is called a nondiametral point in K. Brodskiϊ and Mil'man [2] proved that a space X fails to have normal structure if and only if there is a nonconstant bounded sequence (xn) in X such that the distance from xn+ to the convex hull of {x\,...9xn} tends to the diameter of the set {xk: k e N}; such a sequence is called a diametral sequence in X. One consequence of this fact is that normal structure is a separably-determined property. Several more conditions equivalent to normal structure are given in [5] and in Lemma 3 below. The notation used in this paper is, with perhaps two exceptions, standard. The two exceptions are as follows: if (xn) is a sequence in a