The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

Let B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

We study the minimal length of faithful nuclear representations of operators acting between finite-dimensional Banach spaces and the related minimal number of contact points of the John ellipsoid of maximal volume contained in the unit ball of a finite-dimensional Banach space. In both cases the classical upper estimates, which follow from the Caratheodory theorem, are shown to be exact. Related isometric characterizations of are proved.

J. E. Jayne and C. A. Rogers in [7] introduced the following notion.

Let X be a topological space and p be a metric defined on X × X. X is said to be fragmented by the metric p if, for every ε > 0 and each nonempty subset Y of X there is a nonempty relatively open subset U of Y such that ρ-diam (U)≤ ε.

While investigating Asplund spaces in [15], R. R. Phelps and the author noticed that weak* compact subsets of the duals of Asplund spaces (or equivalently, as it turned out, weak* compact subsets of dual Banach spaces with the Radon-Nikodým property) possessed many properties in common with weakly compact subsets of Banach spaces. The topological study of the spaces homeomorphic to the latter, the so-called Eberlein compact spaces, or EC spaces for short, had flourished and had already yielded a rich collection of results. Therefore it was natural to hope that a similar study of the former might also lead to interesting discoveries. In a series of letters with S. Fitzpatrick exchanged during the summer and the fall of 1981, we started to collect properties of compact spaces that are homeomorphic to weak* compact subsets of the duals of Asplund spaces, which we tentatively called “Asplund compact spaces“. However, as far as we are aware, Reynov's paper [16] is the first study in print of the topological properties of “Asplund compact spaces” or “compacta of RN type” as Reynov termed them.

We first define an inscribed center of a bounded convex body in a normed linear space as the center of a largest open ball contained in it (when such a ball exists). We then show that completeness is a necessary condition for a normed linear space to admit inscribed centers. We show that every weakly compact convex body in a Banach space has at least one inscribed center, and that admitting inscribed centers is a necessary and sufficient condition for reflexivity. We finally apply the concept of inscribed center to prove a type of fixed point theorem and also deduce a proposition concerning so-called Klee caverns in Hilbert spaces.

It is shown that a normed linear space admitting (Chebyshev) centers is complete. Then the ideas in the proof of this fact are used to show that every incomplete CLUR (compactly locally uniformly rotund) normed linear space contains a closed bounded convex subset B with the following properties: (a) B does not contain any farthest point; (b) B does not contain any nearest point (to the elements of its complement).

In our paper [12[ we made extensive use of the details of the proofs given in our earlier paper [11], and, in particular, we claimed that Lemma 3 of [11] holds, not just when Y is a metric space, but also when Y is a Hausdorff space, provided X × Y is a Fréchet space. In a corrigendum to [11], we give a corrected version of this Lemma 3, but it seems to depend, in an essential way, on the assumption that Y is a metric space, or at least a perfectly normal space. In this note we show that a modified version of this Lemma 3 enables us to justify all the theorems in [12] by use of a modified method of selection.

Let A be a commutative Banach algebra with identity of norm 1, X a Banach A-module and G a locally compact abeian group with Haar measure. Then the multipliers from an A -valued function algebra into an X-valued function space is studied. We characterize the multiplier spaces as the following isometrically isomorphic relations under some appropriate conditions:

Si E et F sont deux espaces vectoriels en dualité séparante, M+(E, F) désigne le cône des mesures coniques positives sur E mis en dualité avec F, c'est à dire le cônes des formes postives sur le treillis de fonctions sur E engendré par F. Ce sont des objets plus généraux que les mesures cylindriques admettant des moments finis d'ordre un.

On part d'abord d'une mesure conique représentée par une mesure de Radon sur le complété faible de E et on donne des critéres (par exemple R.N.P.) pour qu'elle le soit sur l'espace E lui-même.

On étudie ensuite les cônes faiblement complets saillants (classe L) contenus dans un espace de Banach ou dans le dual d'un espace de Fréchet F; on montre notamment qu' un cône faiblement fermé contenu dans F′ est dans Lsi son polaire dans F est positivement engendré.

Si B est un espace de Banach et 11 ⊄ B, on cherche à prologner une μ ∈ M+(B′, B) en un élement de M+ (B′, B″). On montre également que, si X est un convexe compact, toute fonction vérifiant le calcul barycentrique sur X est continue sur des ensembles fixes que l'on précise.

Enfin on donne des conditions (de type bornologique) sur un e.l.c.s E, permettant d'interpréter une μ ∈ M+ (E, E′) comme une mesure conique sur un espace normé.

Recently the concept of uniform rotundity was generalized for real Banach spaces by using a type of “area” devised for these spaces. This paper modifies the methods used for uniform rotundity and applies them to weak rotundity in real and complex spaces. This leads to the definition of k-smoothness, k-very smoothness and k-strong smoothness. As an application, several sufficient conditions for reflexivity are obtained.

Some results on fixed points of certain involutions in Banach spaces have been obtained, and whence a few coincidence theorems are also derived. These are indeed generalization of previously known results due to Browder, Goebel-Zlotkiewicz and Iséki. Illustrative examples are also given.

If the second dual of a Banach space E is smooth at each point of a certain norm dense subset, then its first dual admits a long sequence of norm one projections, and these projections have ranges which are suitable for a transfinite induction argument. This leads to the construction of an equivalent locally uniformly rotund norm and a Markuschevich basis for E*.

The structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

Given an ordered Banach space ℬ equipped with an order-norm we construct a larger space ℬ¯ with an order-norm and order-identity such that B is isometrically order-isomorphic to a Banach subspace of B. We also discuss the extension of positive operators from ℬto ℬ¯.

A Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

A subspace of a Banach space is called an operator range if it is the continuous linear image of a Banach space. Operator ranges and operator ideals with fixed range space are investigated. Properties of strictly singular, strictly cosingular, weakly sequentially precompact, and other classes of operators are derived. Perturbation theory and closed semi-Fredholm operators are discussed in the final section.

A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E of Lp and a λ > 1 such that 1E has no λ-unconditional decomposition in terms of rank one operators.

If a Banach space E admits a Markuschevich basis, then E can be renormed to be locally uniformly rotund. When the coefficient space of the basis is 1-norming, and this norm is very smooth, E is weakly compactly generated.

A flat spot in a Banach space X is an element x ∈ Sx = {x ∈ X: ‖x‖ = 1} with the property that the infimum m(x) of the lengths of all curves in Sx joining x to −x is 2. Flat spots occur in every non-superreflexive space when suitably renormed. A study is made of the geometric implications of the existence of flat spots. Connections with other notions such as differentiability, decomposition constants and Kadec-Klee norms are explored and some renorming results for non-superreflexive spaces are proved.