Banach spaces with weak*-sequential dual ball

TL;DR

This paper investigates Banach spaces with weak*-sequential dual balls, establishing conditions under which this property is preserved in subspaces and quotients, and providing examples like Johnson-Lindenstrauss space and certain $C(K)$ spaces.

Contribution

It proves that weak*-sequential dual ball property is stable under certain subspace and quotient conditions, and identifies new examples of such spaces.

Findings

Johnson-Lindenstrauss space $JL_2$ has weak*-sequential dual ball

$C(K)$ spaces for scattered compact $K$ of countable height have weak*-sequential dual ball

The property is preserved under subspaces and quotients with the same property

Abstract

A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball, answering in this way a question of A. Plichko.