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.

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

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    $\begingroup$ Trying to understand the argument. What is the role of $M_i$'s? $\endgroup$ – Bunyamin Sari Nov 18 '19 at 17:52
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    $\begingroup$ Good question, Bunyamin. They are not needed, but they give the lower $c_0$ estimate for linear combinations without stopping to think. $\endgroup$ – Bill Johnson Nov 18 '19 at 18:26
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    $\begingroup$ 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. $\endgroup$ – Spyridon Argyros Nov 24 '19 at 12:12
  • $\begingroup$ You may want to post this as a new question, it is not visible here (e.g., I just saw it). $\endgroup$ – Bunyamin Sari Dec 2 '19 at 16:59

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