Summands in locally almost square and locally octahedral spaces

We study the question whether properties like local/weak almost squareness and local octahedrality pass down from an absolute sum $X\oplus_F Y$ to the summands $X$ and $Y$.Comment: 14 page

A remark on condensation of singularities

Recently Alan D. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I.\,S. G\'al.Comment: 7 page

K\"othe-Bochner spaces and some geometric properties related to rotundity and smoothness

In 2000 Kadets et al. introduced the notions of acs, luacs and uacs spaces, which form common generalisations of well-known rotundity and smoothness properties of Banach spaces. In a recent preprint the author introduced some further related notions and investigated the behaviour of these geometric properties under the formation of absolute sums. This paper is in a sense a continuation of the previous work. Here we will study the behaviour of said properties under the formation of K\"othe-Bochner spaces, thereby generalising some results of Sirotkin on the acs, luacs and uacs properties of $L^p$-Bochner spaces.Comment: 40 pages, 4 figures, partial text overlap with arXiv:1201.230

Some remarks on stronger versions of the Boundary Problem for Banach spaces

Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$ is said to be a boundary for $X$, if every element of $X$ attains its norm on some functional in $B$. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of $X$ which is compact in the topology of pointwise convergence on $B$ is already weakly compact. This problem was recently solved by H.Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.Comment: 15 pages, version 2, references added, two remarks added, some arguments slightly changed, revised version accepted for publication in Applied General Topolog

Rainwater-Simons-type convergence theorems for generalized convergence methods

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.Comment: 10 pages, version 2, references added, one remark added, revised version accepted for publication in Acta et Commentationes Universitatis Tartuensis de Mathematic