Let $M$ be a subspace of $C[0,1]$ isomorphic to $c_0$.

QUESTION: Is it possible to find a normalized disjoint sequence $(f_n)$ in $C[0,1]$ such that the distance of $f_n$ to $M$ tends to $0$ as $n$ goes to $\infty$?

Arguments in favor:

- If $1\leq p<\infty$, then the result is true for subspaces of $L_p(0,1)$ isomorphic to $\ell_p$.
- If we consider the continuous linear projection $P$ on $C[0,1]$ onto $M$, then it was proved by N. Ghoussoub and W.B. Johnson [Math. Z. 194, 153-171 (1987); Theorem I.3] that $P$ is an isomorphism on the closed subspace $[g_n]$ generated by some normalized disjoint sequence $(g_n)$.

With respect to 2, if I were able to show that $[g_n]\cap M$ is infinite dimensional or $[g_n]+M$ is not closed, then it would be easy to get a positive answer.