Topological open problems in the geometry of Banach spaces.
Topological proper separation theorems
Topological Properties of Real Normed Space
In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered...
Topological radicals, I. Basic properties, tensor products and joint quasinilpotence
Topological type of weakly closed subgroups in Banach spaces
The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in which are interesting from the Banach space theory point of view are discussed.
Total Bounded Extensions of Biorthogonal Sequences in Banach Spaces.
Total convexity for powers of the norm in uniformly convex Banach spaces.
Total subspaces in dual Banach spaces which are not norming over any infinite-dimensional subspace
The main result: the dual of separable Banach space X contains a total subspace which is not norming over any infinite-dimensional subspace of X if and only if X has a nonquasireflexive quotient space with a strictly singular quotient mapping.
Tout sous-espace de contient un , d’après D. Aldous
Transformations in the space of almost convergent sequences.
Transitivity of proximinality and norm attaining functionals
We study the question of when the set of norm attaining functionals on a Banach space is a linear space. We show that this property is preserved by factor reflexive proximinal subspaces in spaces and generally by taking quotients by proximinal subspaces. We show, for (ℓ₂) and c₀-direct sums of families of reflexive spaces, the transitivity of proximinality for factor reflexive subspaces. We also investigate the linear structure of the set of norm attaining functionals on hyperplanes of c₀ and...
Transitivity of the norm on Banach spaces.
Translation invariant projections in Sobolev spaces on tori in the L¹ and uniform norms
The idempotent multipliers on Sobolev spaces on the torus in the L¹ and uniform norms are characterized in terms of the coset ring of the dual group of the torus. This result is deduced from a more general theorem concerning certain translation invariant subspaces of vector-valued function spaces on tori.
Translations mesurables et ensembles de Rosenthal
Translations of functions iv vector Hardy classes on the unit disk [Book]
Triebel-Lizorkin spaces with non-doubling measures
Suppose that μ is a Radon measure on , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...
Two geometric constants for operators acting on a separable Banach space.
The main result of this paper is the following: A separable Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence on nested, closed, finite codimensional subspaces with null intersection.
Two mappings related to semi-inner products and their applications in geometry of normed linear spaces
In this paper we introduce two mappings associated with the lower and upper semi-inner product and and with semi-inner products (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.
Two remarks on the Khintchine-Kahane inequality
Two remarks on the structure of sets of exposed and extreme points.