Dual maps and the Dunford-Pettis property

TL;DR

This paper characterizes the points of $w^*$-continuity of dual maps in Banach spaces, linking smooth points, the Schur property, and the Dunford-Pettis property, and explores the continuity properties of dual maps.

Contribution

It establishes a characterization of $w^*$-continuity points of dual maps as smooth points and connects the Schur and Dunford-Pettis properties via dual map continuity.

Findings

Points of $w^*$-continuity are exactly the smooth points.

A Banach space has the Schur property iff it has the Dunford-Pettis property and a sequentially $w$-$w$ continuous dual map at 0.

Existence of smooth Banach spaces with dual maps not $w$-$w$ continuous at 0.

Abstract

We characterize the points of $\left\|\cdot\right\|$-$w^*$ continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially $w$-$w$ continuous at $0$. As consequence, we show the existence of smooth Banach spaces on which the dual map is not $w$-$w$ continuous at $0$.