Rotundity in Lebesgue-Bochner function spaces

Abstract

Abstract.This paper concerns the isometric theory of the LebesgueBochner function space Lp{p, X) where 1 <p < oo. Specifically, the question of whether a geometrical property lifts from X to Lp{ ft, X) is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and fi-convexity. However, it is shown that the RadonRiesz property does not lift from X to Lp( fi, X). Consequently, LebesgueBochner function spaces with the Radon-Riesz property are examined more closely.1. Introduction. Frequently in the literature authors have proved theorems stating that the Banach sequence space lp(X) has a certain geometric property exactly when the Banach spaces X and F have that same property [1], [4], [5], [11], [13], [18], [25]. Less frequently, however, do authors consider the corresponding question for the Lebesgue-Bochner function space Lp( ¡i, X) (see [3] for a discussion of this space). Although most people are willing to believe an affirmative answer to such a question, the proof of such a response is often nontrivial. In 1941, Day [4] proved that Lp(n, X) for 1 <p < oo is uniformly rotund whenever X is uniformly rotund by defining an embedding from the simple functions of Lp( ¡i, X) into the space lp(X) and then applying the corresponding result that he proved for lp(X). Subsequently, McShane [14] gave a direct proof of Day's result. It is well known [5] and much more easily proved that an analogous theorem holds if uniform rotundity is replaced by rotundity (strict convexity).In this paper, theorems of the same ilk are obtained for several convexity properties that lie in strength between uniform rotundity (UR) and rotundity (R) as well as for some other geometrical notions. In particular, if X is locally uniformly rotund (LUR) and 1 <p < oo, it is shown that Lp(p, A") is also locally uniformly rotund. This answers a question posed to us by J. Diestel. Similar theorems are obtained for X weakly uniformly rotund (WUR), uniformly rotund in every direction (URED) or midpoint locally uniformly rotund (MLUR). To avoid confusion, the relative strengths of these distinct rotundity notions are given in the chart below where an arrow denotes implication.Presented to the Society, January 4, 1978; received by the editors January 23, 1978. AMS (MOS) subject classifications (1970). Primary 46B99, 46E40.Key words and phrases. Lebesgue-Bochner function space, locally uniformly rotund, weakly uniformly rotund, S-convex, uniformly non-/'(n), Radon-Riesz property. LUR-h» MLUR /" \ UR R \ / WUR -* URED All of these properties are known to carry over from X to lp(X) for 1 <p < oo. However, neither Day's embedding argument nor McShane's proof for the uniform rotundity case seems adaptable to prove the theorems concerning these generalizations of uniform rotundity in Lp( ¡i, X). Here, a technique is presented that with the appropriate modification of details yields the proofs of all four of the theorems indicated a...

Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

“…We finish the paper by proving that if the norm of X is weakly uniformly rotund (WUR), then L 1 (μ, X ) admits an equivalent norm which is WUR when restricted to any Asplund subspace of L 1 (μ, X ) (Theorem 3.9). We should stress here that, for 1 < p < ∞, the canonical norm of L p (μ, X ) is WUR if the norm of X is WUR, thanks to a result of [29] and the fact that every Banach space admitting a WUR equivalent norm is Asplund (see [19]). …”

P-convexity of Musielak Orlicz function spaces of Bochner type

“…were done by many authors ( see br instance [4], [5], [15], ¡16], [191, [20], [21]). In [12] it is proved that the Orlicz-Bochner function space Ls(g, X) is P-convex uf both Lgg) and X are P-convex.…”

𝑃-convexity of Orlicz-Bochner spaces

“…Although the answer to such a question can often be guessed, the proof of such a response is usually nontrivial. Considerations of that type for various kinds of convexity for L p (µ, X) were carried out by many authors (see for instance [6], [7], [17], [18], [20], [21], [22]). We show that L Φ (µ, X) is P -convex iff both L Φ (µ) and X are P -convex.…”