We discuss renorming properties of the dual of a James tree space JT . We present examples of weakly Lindelöf determined JT such that JT ∗ admits neither strictly convex nor Kadec renorming and of weakly compactly generated JT such that JT ∗ does not admit Kadec renorming although it is strictly convexifiable. The norm of a Banach space is said to be locally uniformly rotund (LUR) if for every ...

We present: i) an example of a Banach space of universal disposition that is not separably injective; ii) an example of a Banach space of universal disposition with respect to finite dimensional polyhedral spaces with the Separable Complementation Property; iii) a new type of space of universal disposition nonisomorphic to the previous existing ones.

We prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal...

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach–Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.

Here we give a necessary and sufficient condition for a Banach space to be separable.

In this chapter, we will consider optimal control problems in function space where we will restrict ourselves to state equations that are both linear in the state and linear in the control: We assume that E is a reflexive, separable Banach space with norm ‖ · ‖E and dual space E∗ and X is another separable Banach space with norm ‖ · ‖X and separable dual space X∗. We further suppose A : D(A) ⊆ ...

The main results of the paper: (1) The dual Banach space X∗ contains a linear subspace A ⊂ X∗ such that the set A of all limits of weak∗ convergent bounded nets in A is a proper norm-dense subset of X∗ if and only if X is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. (2) Let X be a non-reflexive Banach space. Then there exists a convex subse...

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

A Banach space W with a Schauder basis is said to be α-minimal for some α < ω1 if, for any two block subspaces Z,Y ⊆ W, the Bourgain embeddability index of Z into Y is at least α. We prove a dichotomy that characterises when a Banach space has an αminimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characte...