On rough norms on Banach spaces
On scalar-valued nonlinear absolutely summing mappings
We investigate cases ("coincidence situations") in which every scalar-valued continuous n-homogeneous polynomial (or every continuous n-linear mapping) is absolutely (p;q)-summing. We extend some well known coincidence situations and obtain several non-coincidence results, inspired by a linear technique due to Lindenstrauss and Pełczyński.
On Schwartz's C-spaces and Orlicz's O-spaces
On Sectionally Dense Summability Fields.
On self-conjugate Banach spaces
On semi-uniform Kadec-Klee Banach spaces.
On separable Banach spaces containing all separable reflexive Banach spaces
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 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 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.
On sharp triangle inequalities in Banach spaces. II.
On shrinking basic sequences in Banach spaces
On singlevaluedness and (strong) upper semicontinuity of maximal monotone mappings
On smooth points of boundaries of open sets
The notions of smooth points of the boundary of an open set and α(·) intrinsically paraconvex sets are introduced. It is shown that for an α(·) intrinsically paraconvex open set the set of smooth points is a dense -set of the boundary.
On Some Classes of Operators on C(K,X)
Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X) → Y is a strongly bounded operator with representing measure m:Σ → L(X,Y). We show that if T is a strongly bounded operator and T̂:B(K,X) → Y is its extension, then T is limited if and only if its extension T̂ is limited, and that T* is completely continuous (resp. unconditionally...
On some convexity properties in the Besicovitch-Musielak-Orlicz space of almost periodic functions with Luxemburg norm
In this article, it is shown that geometrical properties such as local uniform convexity, mid point local uniform convexity, H-property and uniform convexity in every direction are equivalent in the Besicovitch-Musielak-Orlicz space of almost periodic functions endowed with the Luxemburg norm.
On some convexity properties of generalized Cesàro sequence spaces.
On some convexity properties of Musielak-Orlicz spaces
On some convexity properties of Orlicz spaces of vector valued functions.