Representation of continuous linear functionals on a subspace of a countable Cartesien product of Banach spaces
Representation of the Hausdorff measure of noncompactness in special Banach spaces
Représentations d'opérateurs à valeurs dans L1 (X,..,..).
Resolutions of the identity in certain Banach spaces.
RNP and KMP are equivalent for some Banach spaces with shrinking basis
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.
RNP and KMP are incomparable properties in noncomplete spaces.
Rolle's theorem and negligibility of points in infinite dimensional Banach spaces.
Rotund and uniformly rotund Banach spaces.
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.
Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces
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...
Roughness of two norms on Musielak-Orlicz function spaces
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.
R-Schauder decompositions. Some applications.
Schwartz spaces and compact holomorphic mappings.
Selecting basic sequences in φ-stable Banach spaces
In this paper we make use of a new concept of φ-stability for Banach spaces, where φ is a function. If a Banach space X and the function φ satisfy some natural conditions, then X is saturated with subspaces that are φ-stable (cf. Lemma 2.1 and Corollary 7.8). In a φ-stable Banach space one can easily construct basic sequences which have a property P(φ) defined in terms of φ (cf. Theorem 4.5). This leads us, for appropriate functions φ, to new results on the existence of unconditional...
Semi-groupes holomorphes et -convexité
Semi-groupes holomorphes et -convexité (suite)
Separability of Real Normed Spaces and Its Basic Properties
In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section,...
Separated sequences in asymptotically uniformly convex Banach spaces
We prove that the unit sphere of every infinite-dimensional Banach space X contains an α-separated sequence, for every , where denotes the modulus of asymptotic uniform convexity of X.
Separated sequences in uniformly convex Banach spaces
We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xₙ) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence of (xₙ) such that , where is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space contains...
Separating polynomials on Banach spaces.
In this paper we survey some recent results concerning separating polynomials on real Banach spaces. By this we mean a polynomial which separates the origin from the unit sphere of the space, thus providing an analog of the separating quadratic form on Hilbert space.
Sequential convergences and Dunford-Pettis properties.