My question is a part of an exercise in Banach space theory.
In the space $c_0$, for $n \in \mathbb{N}$, let $s_n=\sum_{j=1}^n e_k=(1,1,\cdots,1,0,\cdots,0)$. It's easy to see that $(s_n)_{n\ge 1}$ is a Schauder basis. I want to find an rearrangement $\sigma:\mathbb N \to \mathbb N$(a bijection, of course) of $(s_n)_{n\ge 1}$ such that $(s_{\sigma(n)})_{n\ge 1}$ is no longer a Schauder basis.
There are some properties for Schauder basis that might be helpful. For instance, if some coordinate functional is not bounded then the rearrangement must not be a Schauder basis. But I have no idea how to find such an rearrangement.
Thanks in advance!
First we identify the biorthogonal functionals: Let $$L_nx=x_n-x_{n+1}.$$Then $L_ns_n=1$, while $L_ns_m=0$ for $m\ne n$.
So if we have $$x=\sum_n a_ns_{\sigma(n)}$$(converging in norm) then $$a_n=L_{\sigma(n)}x.$$Say $$P_nx=\sum_{n=1}^Na_ns_{\sigma(n)} =\sum_{n=1}^NL_{\sigma(n)}xs_{\sigma(n)}.$$If the rearrangement is a Schauder basis we must have $||P_N||$ bounded. But $$(P_Nx)_1=\sum_{n=1}^NL_{\sigma(n)}x,$$so we must also have $||\sum_{n=1}^NL_{\sigma(n)}||$ bounded, and hence $||\sum_{n=M}^NL_{\sigma(n)}||$ must be bounded in $M$ and $N$.
But suppose $\sigma$ is such that $\sigma(M),\sigma(M+1),\dots,\sigma(N)$ is a sequence of even integers. Then there's no cancellation in that last sum; it's easy to see that $||\sum_{n=M}^NL_{\sigma(n)}||=2(N-M+1)$.
So if $\sigma(n)$ contains arbitrarily long sequences of even integers then $(s_{\sigma(n)})$ is not a Schauder basis.