Iwo Labuda

Ideals of Subseries Convergence and Copies of c 0 in Banach Spaces

For a sequence x=(x n ) in a Banach space, define C(x) to be the set of all elements (e n ) of the Cantor cube K={0,1}ℕ for which the series \( \sum\nolimits_{n = 1}^\infty {\varepsilon _n x_n } \) is subseries convergent. The main result of the paper is that a Banach space X contains no isomorphic copy of c 0 if and only if, for every sequence x in X, the set C(x) is F σ in K. A similar equivalence involving ‘ideals of Pettis integrability’ is also shown.

Extensions of Integral Operators

On considere des operateurs integraux non singuliers a noyaux mesurables arbitraires. On etudie des conditions sur des espaces de mesures assurant la completude des domaines propres. On introduit des domaines relatifs, propres et etendus, et on demontre des resultats sur la continuite et la maximalite des extensions des operateurs integraux

Infinite products of filters

Stability of some classes of filters under the (infinite, Tikhonov) product operation is investigated. Applications to productivity of some types of set valued maps are given.

Unity of compactness

Let X be paracompact, that is, every open cover P of X admits a locally finite open refinement R covering X. Does there exist a class D of covers of X such that the paracompactness of X is equivalent to its D-compactness (each D ∈ D admits a finite subcover of X)? We address the question in a more general framework of P/R-compact versus D-compact families of subsets in a pretopological space and obtain an affirmative answer as a corollary.

Relative domains of integral operators

The paper generalizes the known construction of the extended domain of an integral operator relative to an arbitrary range space.The aim of the generalization is to get rid of excessive solidity hypotheses imposed in the previous work on the subject.