On Banach Spaces with the Gelfand-Philips Property.
The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...
Criteria for compactly locally uniformly rotund points in Orlicz spaces are given.
Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that . A sequence (Tₙ)ₙ of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and ℓ₁(Y) if the map defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if . Several results about Taylor coefficients of vector-valued...
We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, does not admit a Schauder basis with constant one.
It is proved that a separable Banach space X admits a representation as a sum (not necessarily direct) of two infinite-codimensional closed subspaces and if and only if it admits a representation as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation such that neither of the operator ranges , contains an infinite-dimensional closed subspace if and only...
We prove that if X is an infinite-dimensional Banach space with smooth partitions of unity then X and X∖ K are diffeomorphic for every weakly compact set K ⊂ X.
In Rolewicz (2002) it was proved that every strongly α(·)-paraconvex function defined on an open convex set in a separable Asplund space is Fréchet differentiable on a residual set. In this paper it is shown that the assumption of separability is not essential.