We investigate rich subspaces of L₁ and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products.
The aim of this paper is to derive some relationships between the concepts of the property of strong introduced recently by Hong-Kun Xu and the so-called characteristic of near convexity defined by Goebel and Sȩkowski. Particularly we provide very simple proof of a result obtained by Hong-Kun Xu.
In the paper [5] L. Drewnowski and the author proved that if is a Banach space containing a copy of then is not complemented in and conjectured that the same result is true if is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if is a finite measure and is a Banach lattice not containing copies of , then is complemented in . Here, we show that the complementability of in together...
In this paper we give estimations of Istratescu measure of noncompactness I(X) of a set X C lp(E1,...,En) in terms of measures I(Xj) (j=1,...,n) of projections Xj of X on Ej. Also a converse problem of finding a set X for which the measure I(X) satisfies the estimations under consideration is considered.
This note is to report some of the advances obtained as a follow-up of the book [2] on the topic of twisted sums of Banach spaces. Since this announcement is no longer enough to contain the theory being developed, we submit the interested reader to [2] and to [1], where full details and proofs shall appear.
It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.
We get a characterization of PCP in Banach spaces with shrinking basis. Also, we prove that the Radon-Nikodym and Krein-Milman properties are equivalent for closed, convex and bounded subsets of some Banach spaces with shrinking basis.
In this paper we prove that the geometrical notions of Rotundity and Uniform Rotundity of the norm in a Banach space are stable for the generalized Banach products.
For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...
In this paper, the criteria of strong roughness, roughness and pointwise roughness of Orlicz norm and Luxemburg norm on Musielak-Orlicz function spaces are obtained.