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We study the Complex Unconditional Metric Approximation Property for translation invariant spaces of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which has (ℂ-UMAP); though these sets are such that contains functions which are not continuous, we show that there is a linear invariant lifting from these spaces into the Baire class 1 functions.
Every separable nonreflexive Banach space admits an equivalent norm such that the set of the weak-extreme points of the unit ball is discrete.
We prove the continuity of the rotundity modulus relative to linear subspaces of normed spaces. As a consequence we reduce the study of uniform rotundity relative to linear subspaces to the study of the same property relative to closed linear subspaces of Banach spaces.
We show that on every nonseparable Banach space which has a fundamental system (e.gȯn every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function such that the set of its Gâteaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function on .
Let be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdorff metric. In the first part of this work we study the density character of and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).
2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12.We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A
showing that inf {max±||1 ± x|| − 1 : x ∈ A, ||x||=ε} ≥ (π/4e)ε²+o(ε²) as ε → 0.
We also give a characterization of two-dimensional subspaces of Banach algebras
containing the identity in terms of polynomial inequalities.
Criteria in order that a Musielak-Orlicz sequence space contains an isomorphic as well as an isomorphically isometric copy of are given. Moreover, it is proved that if , where are defined on a Banach space, does not satisfy the -condition, then the Musielak-Orlicz sequence space of -valued sequences contains an almost isometric copy of . In the case of it is proved also that if contains an isomorphic copy of , then does not satisfy the -condition. These results extend some...
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