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...

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 show the existence of Lipschitz approximable separable spaces which fail Grothendieck's approximation property. This follows from the observation that any separable space with the metric compact approximation property is Lipschitz approximable. Some related results are spelled out.

We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets $K$ and $L$ such that $C\left(L\right)$ is a Lipschitz-quotient of $C\left(K\right)$ (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.

We study the local dual spaces of a Banach space X, which can be described as the subspaces of X* that have the properties that the principle of local reflexivity attributes to X as a subspace of X**. We give several characterizations of local dual spaces, which allow us to show many examples. Moreover, every separable space X has a separable local dual Z, and we can choose Z with the metric approximation property if X has it. We also show that a separable space containing no...

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates.

The aim of this paper is to study the relationships between the concepts of local near uniform smoothness and the properties H and H*.

A new lower bound for the Jung constant $JC\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space $l^{\left(\Phi \right)}$ defined by an N-function Φ is found. It is proved that if $l^{\left(\Phi \right)}$ is reflexive and the function tΦ’(t)/Φ(t) is increasing on $(0,{\Phi }^{-1}\left(1\right)]$, then $JC\left(l^{\left(\Phi \right)}\right)\ge \left({\Phi }^{-1}(1/2)\right)/\left({\Phi }^{-1}\left(1\right)\right)$. Examples in Section 3 show that the above estimate is better than in Zhang’s paper (2003) in some cases and that the results given in Yan’s paper (2004) are not accurate.

Banach spaces which are L-summands in their biduals - for example $l^1$, the predual of any von Neumann algebra, or the dual of the disc algebra - have Pełczyński’s property (V*), which means that, roughly speaking, the space in question is either reflexive or is weakly sequentially complete and contains many complemented copies of $l^1$.

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $𝒦(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $ℒ(X,Y)$ whenever $𝒦(X,X)$ and $𝒦(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $ℒ(X,X)$ and $ℒ(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $𝒦(X,Y)$ in $ℒ(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results...