On nearly uniformly convex and k-uniformly convex spaces

In this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

Denseness of operators which attain their numerical radius

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

On Nonreflexive Banach Spaces Which Contain No c0 or lp

The first infinite-dimensional reflexive Banach space X such that no subspace of X is isomorphic to c0 or lp, 1 ≦ p < ∞, was constructed by Tsirelson [8]. In fact, he showed that there exists a Banach space with an unconditional basis which contains no subsymmetric basic sequence and which contains no superreflexive subspace. Subsequently, Figiel and Johnson [4] gave an analytical description of the conjugate space T of Tsirelson's example and showed that there exists a reflexive Banach space with a symmetric basis which contains no superreflexive subspace; a uniformly convex space with a symmetric basis which contains no isomorphic copy of lp, 1 < p < ∞; and a uniformly convex space which contains no subsymmetric basic sequence and hence contains no isomorphic copy of lp, 1 < p < ∞. Recently, Altshuler [2] showed that there is a reflexive Banach space with a symmetric basis which has a unique symmetric basic sequence up to equivalence and which contains no isomorphic copy of lp, 1 < p < ∞.

Uniformly Lipschitzian Families of Transformations in Banach Spaces

The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ Kwhere {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)