We prove that an Orlicz space equipped with the Luxemburg norm has uniformly normal structure if and only if it is reflexive.

In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].

We show that when the conjugate of an Orlicz function ϕ satisfies the growth condition Δ⁰, then the reflexive subspaces of $L^{\varphi }$ are closed in the L¹-norm. For that purpose, we use (and give a new proof of) a result of J. Alexopoulos saying that weakly compact subsets of such $L^{\varphi }$ have equi-absolutely continuous norm.

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded closed convex...

For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...