A note on interpolation and higher integrability.
We provide a corrected proof of [1, Théorème 9] stating that any metrizable infinite-dimensional simplex is affinely homeomorphic to the intersection of a decreasing sequence of Bauer simplices.
It is shown that every strongly lattice norm on can be approximated by smooth norms. We also show that there is no lattice and Gâteaux differentiable norm on .
The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between L-Dunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.
We show that the following well-known open problems on existence of Lipschitz isomorphisms between subsets of Hilbert spaces are equivalent: Are balls isomorphic to spheres? Is the whole space isomorphic to the half space?
Let be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of , the algebra of all bounded linear operators on a Hilbert space , is an automorphism.
Answering a question of Pisier, posed in [10], we construct an L-set which is not a finite union of translates of free sets.