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 $su{p}_{\left|z\right|<1}(1-|z\left|²\right)\left|\right|f^{\text{'}}\left(z\right)\left|\right|<\infty$. 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 ${\sum }_{n=0}^{\infty }xₙzⁿ\to \left(Tₙ\left(xₙ\right)\right)ₙ$ 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 $su{p}_k{\sum }_{n=2^k}^{2^{k+1}}\left|\right|Tₙ\left|\right|²<\infty$. Several results about Taylor coefficients of vector-valued...

We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_p\left(\right)$ 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, $H_p$ does not admit a Schauder basis with constant one.

It is proved that a separable Banach space X admits a representation $X=X_1+X_2$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_1$ and $X_2$ if and only if it admits a representation $X=A_1\left(Y_1\right)+A_2\left(Y_2\right)$ 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 $X=T_1\left(Z_1\right)+T_2\left(Z_2\right)$ such that neither of the operator ranges $T_1\left(Z_1\right)$, $T_2\left(Z_2\right)$ contains an infinite-dimensional closed subspace if and only...

We prove that if X is an infinite-dimensional Banach space with $C^p$ smooth partitions of unity then X and X∖ K are $C^p$ 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.