Smooth renormings of the Lebesgue–Bochner function space L¹(μ,X)

Abstract

We show that, if µ is a probability measure and X is a Banach space, then the space L 1 (µ, X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L 1 (µ, X) has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.

Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

“…II, Corollary 3.3]). It follows that the aforementioned result in [13] applies exactly to Asplund subspaces Y ⊆ L 1 (μ, X ). Bearing in mind that M(t) = |t| − log(1 + |t|) also fulfills the requirements of Theorem 3.9, we deduce that if the norm · of X is Fréchet smooth and WUR, then L 1 (μ, X ) admits an equivalent norm which is Fréchet smooth and WUR when restricted to any Asplund subspace Y ⊆ L 1 (μ, X ).…”

“…Some ideas from [23] were adapted to the vector-valued case in [13,14] to show that some convexity and smoothness properties of X lift to L 1 (μ, X ) when equipped with the Orlicz-type equivalent norm ||| · ||| defined below.…”

Differentiability of $$L^p$$ L p of a vector measure and applications to the Bishop–Phelps–Bollobás property

“…The proof of this lemma is based on the fact that if the space E(µ) is order continuous, then the (topological) dual space E(µ) * coincides isometrically with the Köthe dual space E(µ) , which can be identified with the set of functionals defined by integrals given by a function (see [23,Proposition 2.16,Remark 3.8] and the references therein), and a Šmulyan type criterion for uniformly Gâteaux smoothness (see e.g. [15,Lemma 2.5]), which ensures that a Banach space X is uniformly Gâteaux smooth if, and only if, X is Gâteaux smooth and for every h ∈ B X , the mapping S X x −→ • (x)(h) ∈ R is uniformly continuous.…”