The distance $dist (\mathcal{B},X)$ when $\mathcal{B}$ is a boundary of $B(X^{**})$

Abstract

Abstract. Let X be a real Banach space and let B be a boundary of the unit ball B(X * *

Measures of weak non-compactness in spaces of nuclear operators

“…Another approach was used to prove a quantitative version of the Krein theorem -it was done independently in three papers [11,13,8] using different methods. The second approach inspired a fruitful research, the applications include quantitative versions of several classical theorems on weak compactness (Eberlein-Šmulyan theorem [2], Gantmacher theorem [3], James compactness theorem [7,14]), a characterization of subspaces of weakly compactly generated spaces [12] or a quantitative view on several properties of Banach spaces (Dunford-Pettis property [17], reciprocal Dunford-Pettis property [19], Banach-Saks property [5] etc. ).…”

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

“…They are used to prove more precise versions of known results and to establish new results as well. As an illustration we mention a fixed-point theorem [21], quantitative versions of Krein's theorem [26,31], James' compactness theorem [18,32], Eberlein-Šmulyan theorem [3] and Gantmacher's theorem [4].…”

Quantification of Pełczyński's property (V)

“…Our inspiration comes from plenty of recently published quantitative results. Let us mention for example quantitative versions of Krein's theorem , the Eberlein–Šmulyan and the Gantmacher theorem , James' compactness theorem , weak sequential continuity and the Schur property , the Dunford–Pettis and the reciprocal Dunford–Pettis property , the Grothendieck property , and the Banach‐Saks property . Quantitative approach contributes to deeper understanding of the notions and properties in question, in some cases substantially (see e.g.…”