Related papers: Weak$^*$-sequential properties of Johnson-Lindenst…

A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is…

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 prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology…

In this paper, we introduce the notions uniformly p-convergent sets and weakly p-sequentially continuous differentiable mappings. Then we obtain a sufficient condition for those Banach spaces which either contain no copy of $\ell_1$ or have…

E. Oja, T. Viil, and D. Werner showed, in [Totally smooth renormings, Archiv der Mathematik, 112, 3, (2019), 269--281] that a weakly compactly generated Banach space $(X,\|\cdot \|)$ with the property that every linear functional on $X$ has…

A Banach space $X$ has the ball fixed point property (BFPP) if for every closed ball $B$ and for every nonexpansive mapping $T\colon B\to B$, there is a fixed point. We study the BFPP for $C(K)$-spaces. Our goal is to determine topological…

Let X and Y be Banach spaces and F a subset of B_{Y^*}. Endow Y with the topology \tau_F of pointwise convergence on F. Let T: X^* \to Y be a bounded linear operator which is (w^*, \tau_F) continuous. Assume that every vector in the range…

The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G.…

A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T: X -> Y is weakly compact. H. Pfitzner proved that C*-algebras have Pelczynski's property (V). In the preprint "H.…

We investigate the weak Banach--Saks property in the setting of H\"older spaces over metric spaces. We show that, for every infinite metric space $(M,d)$ and every $\alpha \in (0,1]$, the H\"older space $C^{\alpha}(M)$ fails to have the…

We consider the class of quantum mechanical master equations defined on a generic Banach space, arising by projecting weakly perturbed one-parameter groups of isometries. We show that the possible semigroup approximations are far from…

We prove that there exist Banach spaces not containing $\ell_1$, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the…

We have recently introduced the trimming property for a complete Riemannian manifold $N^{n}$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1, p}(B^m; N^{n})$ when $p \in \{1, \dotsc, m\}$. We prove in…

Letting $E$, $F$ be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator $E\rightarrow F$ is unconditionally converging, then every polynomial from $E$ to $F$ is unconditionally…

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…

One shows for Banach bundles in a certain class that having a second countable locally compact Hausdorff base space and separable fibers implies the separability of the Banach space of the all sections that vanish at infinity. In the…

We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over $\mathbb{F}$ with a certain…

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is…

We prove that for every $n\in \mathbb{N}$ there exists a metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$, a Banach space $(Z,\|\cdot\|_Z)$ and a $1$-Lipschitz function $f:S\to Z$ such that the Lipschitz constant of every function…