We prove that the topology of the additive group of the Banach space c0 is not induced by weakly almost periodic functions or, which is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. This answers, on the negative, questions of Megrelishvili [14] and Glasner and Megrelishvili [7]. We show, in contrast, that additive groups of Schwartz locall...

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex.

The main result is that a separable Banach space with the weak∗ unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with an unconditional basis iff it is isomorphic to a quotient of a space with an unconditional basis, which solves...

and Applied Analysis 3 Remark 1.1. If E is a reflexive strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0 then x y. From 1.6 , we have ‖x‖ ‖y‖. This implies 〈x, Jy〉 ‖x‖2 ‖Jy‖2. From the definitions of j, we have Jx Jy, that is, x y, see 8, 9 for more details. Let C be a nonempty closed convex subset of a smooth Bana...

A generalization of Phelps’ lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland’s principle and Danes̆’ drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997. We show...

The convex feasibility problem CFP of finding a point in the nonempty intersection ⋂N i 1Ci is considered, where N 1 is an integer and the Ci’s are assumed to be convex closed subsets of a Banach space E. By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for s...

In this paper, we show that a closed convex set C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonemp...

V. D. Milman proved in [14] that the product of two strictly singular operators on L p [0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying ...

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [17]. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of multivalued maps has applications in control theory, convex optimization, differential inclusions, and economics (see, e.g., [3, 8, 16, 22]). The theory of multivalu...