# Copies of $c_0$ in $C[0,1]$ and disjoint sequences

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:

1. If $$1\leq p<\infty$$, then the result is true for subspaces of $$L_p(0,1)$$ isomorphic to $$\ell_p$$.
2. 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.

• What do you mean with a "normalized disjoint sequence"? – Wojowu Nov 16 '19 at 12:18
• A sequence $(f_n)$ such that $\|f_n\|_\infty=1$ for all $n$ and $f_k\cdot f_l=0$ for $k\neq l$. – M.González Nov 16 '19 at 12:43

Take sequences of clopen sets $$M_i$$, $$N_i$$, s.t. any two of the $$N_i$$ have non empty intersection but any three have empty intersection, and the $$M_i$$ are pairwise disjoint and disjoint from the $$N_i$$. Let $$f_i$$ be the characteristic function of $$M_i \cup N_i$$. This gives a counterexample in $$C(\Delta)$$, $$\Delta$$ a Cantor set. Doing it in $$C[0,1]$$ is similar but not as nice to describe.
• Trying to understand the argument. What is the role of $M_i$'s? – Bunyamin Sari Nov 18 '19 at 17:52
• Good question, Bunyamin. They are not needed, but they give the lower $c_0$ estimate for linear combinations without stopping to think. – Bill Johnson Nov 18 '19 at 18:26
• I wonder if Bill's answer leads to a positive answer to the following reformulation of the above question. For a given n consider the closed subset $K_n$ of the Cantor set consisting of all characteristic functions of F with #F < n+1.For all n the projections $p_k$ are equivalent to $c_0$ basis. For n=1 they are isometric to the $c_0$ basis and for n=2 is Bill's example.My question is if the subspace M, in the question, can be approximated by a sequence ($g_m$) isometric to one of the above described sequences. – Spyridon Argyros Nov 24 '19 at 12:12