Note on extreme points in Marcinkiewicz function spaces.
Note on operational quantities and Mil'man isometry spectrum.
Let X and Y be infinite dimensional Banach spaces and let L(X,Y) be the class of all (linear continuous) operators acting between X and Y. Mil'man [5] introduced the isometry spectrum I(T) of T ∈ L(X,Y) in the following way:I(T) = {α ≥ 0: ∀ ε > 0, ∃M ∈ S∞(X), ∀x ∈ SM, | ||Tx|| - α | < ε}},where S∞(X) is the set of all infinite dimensional closed subspaces of X and SM := {x ∈ M: ||x|| = 1} is the unit sphere of M ∈ S∞(X). (...)
Note on separation of convex sets
Nouvelles classes d'espaces de Banach à prédual unique
Nouvelles propriétés des espaces et
Numerical index of vector-valued function spaces
We show that the numerical index of a -, -, or -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.
Numerical index with respect to an operator
We introduce new concepts of numerical range and numerical radius of one operator with respect to another one, which generalize in a natural way the known concepts of numerical range and numerical radius. We study basic properties of these new concepts and present some examples.
Obstacles to bounded recovery.
On a class of Hausdorff compact and GSG Banach spaces
On a conjecture of L. Veselý
On a dual locally uniformly rotund norm on a dual Vašák space
We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.
On a Property of the Norm which is Close to Local Uniform Rotundity.
On a recent result of G. Pisier
On a result of J. Johnson
On almost everywhere differentiability of the metric projection on closed sets in ,
Let be a closed subset of and let denote the metric projection (closest point mapping) of onto in -norm. A classical result of Asplund states that is (Fréchet) differentiable almost everywhere (a.e.) in in the Euclidean case . We consider the case and prove that the th component of is differentiable a.e. if and satisfies Hölder condition of order if .
On an extension of norms from a subspace to the whole Banach space keeping their rotundity
Let ℛ denote some kind of rotundity, e.g., the uniform rotundity. Let X admit an ℛ-norm and let Y be a reflexive subspace of X with some ℛ-norm ∥·∥. Then we are able to extend ∥·∥ from Y to an ℛ-norm on X.
On an -Birkhoff orthogonality.
On areas and integral geometry in Minkowski spaces.
On Asplund functions
A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.
On Averaging Null Sequences of Real-Valued Functions
If ξ is a countable ordinal and (fk) a sequence of real-valued functions we define the repeated averages of order ξ of (fk). By using a partition theorem of Nash-Williams for families of finite subsets of positive integers it is proved that if ξ is a countable ordinal then every sequence (fk) of real-valued functions has a subsequence (f'k) such that either every sequence of repeated averages of order ξ of (f'k) converges uniformly to zero or no sequence of repeated averages of order ξ of (f'k)...