Related papers: Weak compactness and Orlicz spaces

Let $\Ps(\N)$ be the set of all finite subsets of $\N$, endowed with the product topology. A description of the compact subsets of $\Ps(\N)$ is given. Two applications of this result to Banach space theory are shown : (1) a characterization…

For $0\leqslant \xi\leqslant \omega_1$, we define the notion of $\xi$-weakly precompact and $\xi$-weakly compact sets in Banach spaces and prove that a set is $\xi$-weakly precompact if and only if its weak closure is $\xi$-weakly compact.…

We provide an alternative proof of the theorem saying that any Va\v{s}\'ak (or, weakly countably determined) Banach space admits a full $1$-projectional skeleton. The proof is done with the use of the method of elementary submodels and is…

An Orlicz space $L^{\Phi}(\Omega)$ is a Banach function space defined by using a Young function $\Phi$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^{\Phi}([0,1])$, a locally compact second countable…

The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the closed convex…

We characterize the relatively compact subsets of $L^1\left(\| m \| \right),$ the quasi-Banach function space associated to the semivariation of a given vector measure $m$ showing that the strong connection between compactness, uniform…

This article is a follow-up to arXiv:2304.04373. We establish necessary and sufficient conditions for weighted Orlicz-Poincar\'e inequalities in product spaces. These results follow the work of Chua and Wheeden, who established similar…

We improve our results on boundedness of the Riesz potential in the central Morrey-Orlicz spaces and the corresponding weak-type version. We also present two new properties of the central Morrey-Orlicz spaces: nontriviality and inclusion…

Given a real Banach space $\mathcal{X}$ and probability space $(\Omega, \Sigma, \mu)$ we characterize the countable additivity of Henstock-Dunford integral for Henstock integrable function taking values in $X$ as those weakly measurable…

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the…

We prove that given a locally compact Hausdorff space, $K$, and a compact C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies that every convex combination of slices of the closed unit ball is relatively weakly…

We give a characterization of the existence of copies of $c_{0}$ in Banach spaces in terms of indexes. As an application, we deduce new proofs of James Distortion theorem and Bessaga-Pe{\l}czynski theorem about weakly unconditionally Cauchy…

It is shown that if the dual of a separable Banach space has Property($K^*$) then the original space has the weak fixed point property. This is an improvement of previously results.

In this paper, several weak Orlicz-Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient…

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we…

It is shown that the weak $L^p$ spaces $\ell^{p,\infty}, L^{p,\infty}[0,1]$, and $L^{p,\infty}[0,\infty)$ are isomorphic as Banach spaces.

We introduce the concept of weak L-P property for a pair of Banach spaces, in the study of extreme contractions. We give examples of pairs of Banach spaces (not) satisfying weak L-P property and apply the concept to compute the exact number…

Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let $\Phi$ be a Young function. In this paper, we consider the Orlicz space $L^\Phi(G)$ and investigate its algebraic property under the twisted…

Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space $E$ and its closed unit ball $B$, both endowed with the weak topology of $E$. We show that $B$ has the Pytkeev property if and only if…

We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be…