A counterexample to a question of R. Haydon, E. Odell and H. Rosenthal

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 $n \in \mathbf {N}$ 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 the negative a question asked by H. Haydon, E. Odell and H. Rosenthal.