Norming points and unique minimality of orthogonal projections.
Norms for copulas.
Norms in product spaces which preserve approximation properties.
Norms of hypercyclic sequences.
Norms on super-reflexive Banach spaces
Norms that locally depend on countably many linear functionals.
Note on a theorem of S. Mercourakis about weakly -analytic Banach spaces
Note on bi-Lipschitz embeddings into normed spaces
Let , be metric spaces and an injective mapping. We put , and (the distortion of the mapping ). We investigate the minimum dimension such that every -point metric space can be embedded into the space with a prescribed distortion . We obtain that this is possible for , where is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into are obtained by a similar method.
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.