We prove that the topology of the additive group of the Banach space c₀ is not induced by weakly almost periodic functions or, what is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. We show, in contrast, that additive groups of Schwartz locally convex spaces are always representable as groups of isometries on some reflexive Banach space.
We show that for "most" compact nonmetrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact nonmetrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.
We obtain the equivalence of the properties and (NUC) in Orlicz function spaces. This answers a question raised by Y. Cui, R. Pluciennik and T. Wang.
In reflexive Banach spaces with some degree of uniform convexity, we obtain estimates for Kottman's separation constant in terms of the corresponding modulus.
Let W and L be complementary subspaces of a Banach space X and let P(W,L) denote the projection on W along L. We obtain a sufficient condition for a subspace M of X to be complementary to W and we derive estimates for the norm of P(W,L) - P(W,M).
The purpose of this paper is to continue the investigations on extremal values for inner and outer radii of the unit ball of a finite-dimensional real Banach space for the Holmes-Thompson and Busemann measures. Furthermore, we give a related new characterization of ellipsoids in via codimensional cross-section measures.
For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index which is universal for the class of separable Banach spaces with Szlenk index not exceeding . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.
We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Kashin decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's position with a Euclidean ball of a given radius, which leads to a new and simplified proof of the randomized isomorphic Dvoretzky theorem. Our results also include a characterization of spaces with nontrivial cotype in terms of arrangements of Euclidean subspaces.