On second derivates of convex functions.
On Sectionally Dense Summability Fields.
On Segal's postulates for general quantum mechanics
On self-conjugate Banach spaces
On semi-inner product spaces, I
On semi-inner product spaces, II
On semi-uniform Kadec-Klee Banach spaces.
On separable Banach spaces containing all separable reflexive Banach spaces
On separation theorems for subadditive and superadditive functionals
We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between...
On sequences in vector lattices
In this paper we investigate conditions for a system of sequences of elements of a vector lattice; analogous conditions for systems of sequences of reals were studied by D. E. Peek.
On sequential convergence in weakly compact subsets of Banach spaces
We construct an example of a Banach space E such that every weakly compact subset of E is bisequential and E contains a weakly compact subset which cannot be embedded in a Hilbert space equipped with the weak topology. This answers a question of Nyikos.
On sequential properties of Banach spaces, spaces of measures and densities
We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space can be naturally expressed in terms of weak* continuity of seminorms on the unit ball of . We attempt to carry out a construction of a Banach space of the form which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the weak* sequential closure of atomic measures, and the set-theoretic properties of generalized...
On sequentially M-integrable distributions
On sequentially retractive inductive limits.
On sets always of the first category
On sets of discontinuities of functions continuous on all lines
Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a -smooth function on and a closed set nowhere dense in such that there does not exist any linearly continuous function on (i.e., function continuous on all lines) which is discontinuous at each point of . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our...
On sets of non-differentiability of Lipschitz and convex functions
We observe that each set from the system (or even ) is -null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on is -strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex...
On sets of small measure
On set-valued cone absolutely summing maps
Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of , and to derive necessary...
On sharp reiteration theorems and weighted norm inequalities
We prove sharp end forms of Holmstedt's reiteration theorem which are closely connected with a general form of Gehring's Lemma. Reverse type conditions for the Hardy-Littlewood-Pólya order are considered and the maximal elements are shown to satisfy generalized Gehring conditions. The methods we use are elementary and based on variants of reverse Hardy inequalities for monotone functions.