Related papers: Weak* sequential closures in Banach space theory a…

Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences…

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator…

We investigate the question whether the (I)-envelope of any subset of a dual to a Banach space $X$ may be described as the closed convex hull in a suitable topology. If $X$ contains no copy of $\ell^1$ then the weak topology generated by…

The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction…

Let $X_1, \ldots, X_m$ be Banach spaces and let $E_1, \ldots, E_m,F$ be Banach lattices. Our main results read as follows: (i) The linear adjoint $A^*$ of a continuous multilinear operator $A \colon X_1 \times \cdots \times X_m \to F$ is…

We study long chains of iterated weak* derived sets, that is sets of all weak* limits of bounded nets, of subspaces with the additional property that the penultimate weak* derived set is a proper norm dense subspace of the dual. We extend…

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such…

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional…

A Banach space $X$ is said to have property ($\mu^s$) if every weak$^*$-null sequence in $X^*$ admits a subsequence such that all of its subsequences are Ces\`{a}ro convergent to $0$ with respect to the Mackey topology. This is stronger…

This paper deals with the interplay of the geometry of the norm and the weak topology in Banach spaces. Both dual and intrinsic connections between weak forms of rotundity and smoothness ared discussed. Weakly exposed points, weakly locally…

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an…

Using a strengthening of the concept of $\K$ set, introduced in this paper, we study a certain subclass of the class of $\K$ Banach spaces; the so called strongly $\K$ Banach spaces. This class of spaces includes subspaces of strongly…

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$…

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.…

The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use \textit{ad hoc} arguments and few general techniques are known. Motivated by the…

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear…

We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following…

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We…

We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space $E$ with a Banach predual. We define the…