Let 𝓐 be a compatible collection of bounded subsets in a normed linear space. We give a characterization of the following generalized Mazur intersection property: every closed convex set A ∈ 𝓐 is an intersection of balls.
Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space means that, for every element in the unit sphere of , we have We note that, in general, the property of convex transitivity for a Banach...
For every element x** in the double dual of a separable Banach space X there exists the sequence of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class (resp. to the class ) as the elements with the sequence equivalent to the usual basis of (resp. as the elements with the sequence equivalent to the usual basis...
We apply the Chebyshev coefficients λf and λb, recently introduced by the authors, to obtain some results related to certain geometric properties of Banach spaces. We prove that a real normed space E is an L1-predual if and only if λf(E) = 1/2, and that if a (real or complex) normed space E is a P1 space, then λb(E) equals λb(K), where K is the ground field of E.
Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with -base and data in , a subspace of 1. We derive our results, considering the action of an adjoint operator on , a predual of , and using known properties of this last space.
A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact....