We prove that if X is a compact topological space which contains a nontrivial metrizable connected closed subset, then the vector lattice C(X) does not carry any sygma-Lebesgue topology.

We shall prove the following statements: Given a sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ in a Banach space $𝐗$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) ${\left\{b_n\right\}}_{n=1}^{\infty }$ of the sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ such that $$\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^n{b}_j=a$$ whenever $a$ belongs to the closed convex hull of the set of weak limit points of ${\left\{a_n\right\}}_{n=1}^{\infty }$. In case $𝐗$ has the Banach-Saks property and ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the...

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$, $1\le p<\infty$.

We study limiting K- and J-methods for arbitrary Banach couples. They are related by duality and they extend the methods already known in the ordered case. We investigate the behaviour of compact operators and we also discuss the representation of the methods by means of the corresponding dual functional. Finally, some examples of limiting function spaces are given.

This article is divided into two parts. The first one is on the linear structure of the set of norm-attaining functionals on a Banach space. We prove that every Banach space that admits an infinite-dimensional separable quotient can be equivalently renormed so that the set of norm-attaining functionals contains an infinite-dimensional vector subspace. This partially solves a question proposed by Aron and Gurariy. The second part is on the linear structure of dominated operators. We show that the...

Linear topological properties of the Lumer-Smirnov class $L{N}_{\ast }{(}^n)$ of the unit polydisc ${}^n$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $L{N}_{\ast }{(}^n)$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $L{N}_{\ast }{(}^n)$.

Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and $T:\overline{span}\left(f\left(X\right)\right)\to X$ be the Figiel operator with $T\circ f=I{d}_X$ and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when $\overline{span}\left(f\left(X\right)\right)$ is weakly nearly strictly convex.

We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a...