If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂({\overline{co}}^{w*}\left(K\right),C):=supinf\left|\right|k-c\left|\right|:c\in C:k\in {\overline{co}}^{w*}\left(K\right)$ from ${\overline{co}}^{w*}\left(K\right)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂({\overline{co}}^{w*}\left(K\right),C)\le Md̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...

The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], ${\theta }_{n+m}\ge \theta ₙ\theta ₘ$ and $li{m}_n\theta {ₙ}^{1/n}=1$, admits an $\ell {₁}^{\omega }$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $li{m}_n\theta {ₙ}^{1/n}=1$, such that, for every n ∈ ℕ, $\left|\right|{\sum }_{k=1}^m{x}_k\left|\right|\ge \theta ₙ{\sum }_{k=1}^m\left|\right|x_k\left|\right|$ for every ₙ-admissible block sequence ${\left(x_k\right)}_{k=1}^m$ of vectors in X, then there exists c > 0 such...

An approximation property of divergent sequences in normed vector spaces is discussed.

Given a positive Banach-Saks operator T between two Banach lattices E and F, we give sufficient conditions on E and F in order to ensure that every positive operator dominated by T is Banach-Saks. A counterexample is also given when these conditions are dropped. Moreover, we deduce a characterization of the Banach-Saks property in Banach lattices in terms of disjointness.

We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented.

This paper is an extension of the work done in [Morsli M., Bedouhene F., Boulahia F., Duality properties and Riesz representation theorem in the Besicovitch-Orlicz space of almost periodic functions, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 103--117] to the Besicovitch-Musielak-Orlicz space of almost periodic functions. Necessary and sufficient conditions for the reflexivity of this space are given. A Riesz type ``duality representation theorem'' is also stated.

In [6], the classical Riesz representation theorem is extended to the class of Besicovitch space of almost periodic functions $B^q$ a.p., $q\in ]1,+\infty [$ . It is also shown that this space is reflexive. We shall consider here such results in the context of Orlicz spaces, namely in the class of Besicovitch-Orlicz space of almost periodic functions $B^{\phi }$ a.p., where $\phi$ is an Orlicz function.

Several properties of weakly p-summable sequences and of the scale of p-converging operators (i.e., operators transforming weakly p-summable sequences into convergent sequences) in projective and natural tensor products with an lp space are considered. The last section studies the Dunford-Pettis property of order p (i.e., every weakly compact operator is p-convergent) in those spaces.