We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited${}^{*}$ and the $p$-(SR${}^{*}$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited${}^{*}$ and $p$-Right${}^{*}$ subsets. The $p$-$L$-limited${}^{*}$ property is studied in some spaces of operators.

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...