On the -angle and -angle in normed spaces.
On the Banach-Mazur distance of finite-dimensional symmetric Banach spaces and the hypergeometric distribution
On the Banach-Stone problem
Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of then T can be written as a weighted composition...
On the Bishop-Phelps-Bollobás theorem for operators and numerical radius
We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and -sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ)...
On the boundary of a polyhedral Banach space.
On the Busemann area in Minkowski spaces.
On the calculation of the Dunkl-Williams constant of normed linear spaces
Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.
On the C-farthest points.
On the Chebyshev alternation theorem.
On the classes of hereditarily Banach spaces
Let denote a specific space of the class of Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily Banach spaces. We show that for the Banach space contains asymptotically isometric copies of . It is known that any member of the class is a dual space. We show that the predual of contains isometric copies of where . For it is known that the predual of the Banach space contains asymptotically isometric copies of . Here we...
On the compact approximation property
We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net of compact operators on X such that and in the strong operator topology. Similar results for dual spaces are also proved.
On the complexity of Hamel bases of infinite-dimensional Banach spaces
We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.
On the complexity of some classes of Banach spaces and non-universality
These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded operators,...
On the diameter of the Banach-Mazur set
On every subspace of which contains an uncountable -independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of is infinite. This provides a partial answer to a question asked by Johnson and Odell.
On the differentiation of convex functions
On the distance between two Chebyshev sets in Banach spaces.
On the distribution of the norm for a gaussian measure
On the equality of injective and projective tensor products
On the exact value of Jung constants of Orlicz sequence spaces.
On the existence of almost greedy bases in Banach spaces
We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an...