In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $\sigma$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszy\'nski and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of $ZFC + \neg CH$. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.
We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal $C_0$-semigroup has an unbounded generator....
G. Ara'ujo, Mateus Azevedo, Flank Bezerra et al.· 0 citations
The main purpose of this paper is to investigate the relationships between some classes of non-Archimedean injective Banach spaces. The results obtained reveal sharp and interesting contrasts with the classical situation (i.e. for Banach spaces over the reals
\mathbb{R}
R
or the complex numbers...
C. Pérez-García· Journal of Convex Analysis· 1 citation
We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $\sigma$-complete vector lattice endowed with...
It is known that the free $p$-convex Banach lattice on a Banach space $X$ can be represented as a space of functions on the unit ball of $X^*$. In this way, it gives rise to certain related (larger) lattices. To investigate such lattices, we introduce a new tool, related to operators into $X$. This tool is then used to...
The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon--Nikod\'ym property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admit...
In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. In particular, we show that a normed lattice $F$ is order continuous iff ev...
Eugene Bilokopytov· 0 citations
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