A counterexample to a question of Haydon, Odell and Rosenthal

George Androulakis

paper

A counterexample to a question of Haydon, Odell and Rosenthal

arXiv:math/9610213

Abstract

We give an example of a compact metric space $K$ , an open dense subset $U$ of $K$ , and a sequence $(f_{n})$ in $C (K)$ which is pointwise convergent to a non-continuous function on $K$ , such that for every $u \in U$ there exists $n \in N$ with $f_{n} (u) = f_{m} (u)$ for all $m \geq n$ , yet $(f_{n})$ is equivalent to the unit vector basis of the James quasi-reflexive space of order 1. Thus $c_{0}$ does not embed isomorphically in the closed linear span $[f_{n}]$ of $(f_{n})$ . This answers in negative a question asked by H. Haydon, E. Odell and H. Rosenthal.