$c\_{0}$, the space of the scalar sequence that converges to $0$ endowed with the sup norm, has two well-known bases: the unit vector basis $(e\_{n})\_{n}$, where $e\_{n}(k)=1$ if $k=n$ and $0$ otherwise, and the summing basis $(s\_{n})\_{n}$, where $s\_{n}=\sum\_{k=1}^{n}e\_{k}$. It is known that $c\_{0}$ has no boundedly complete basis. Recall that a basis $(x\_{n})\_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a\_{n})\_{n=1}^{\infty}$ with $\sup\_{n}\|\sum\_{i=1}^{n}a\_{i}x\_{i}\|<\infty$, the series $\sum\_{n=1}^{\infty}a\_{n}x\_{n}$ converges.


We introduce a quantity measuring (non-)bounded completeness of a basis as follows:


Let $(x\_{n})\_{n=1}^\infty$ be a bounded sequence in a Banach space $X$. We set $$\textrm{ca}((x\_{n})\_{n=1}^\infty)=\inf\_{n}\sup\_{k,l\geq n}\|x\_{k}-x\_{l}\|.$$ Then $(x\_{n})\_{n=1}^\infty$ is norm-Cauchy if and only if $\textrm{ca}((x\_{n})\_{n=1}^\infty)=0$.


Let $(x\_{n})\_{n=1}^\infty$ be a basis for a Banach space $X$. We set $$\textrm{bc}((x\_{n})\_{n=1}^\infty)=\sup\Big\{\textrm{ca}((\sum\_{i=1}^{n}a\_{i}x\_{i})\_{n=1}^\infty)\colon
(\sum\_{i=1}^{n}a\_{i}x\_{i})\_{n=1}^\infty\subseteq B\_{X}\Big\},$$ where $B\_{X}$ is the closed unit ball of $X$.


Since we have proved that $\operatorname{bc}((e\_{n})\_{n})=\operatorname{bc}((s\_{n})\_{n})=1$, we have the following natural question:


Question. $\textrm{bc}((x\_{n})\_{n=1}^\infty)=1$ for every basis $(x\_{n})\_{n}$ in $c\_{0}$ ?


To answer the question, I want to know if there are other bases in $c\_{0}$ besides these two important classes of bases. Are there more references about bases in $c\_{0}$? Thank you.



Bases in $c_{0}$
You have answered your own question modulo a lemma that is basically obvious:


Lemma. Let $(x\_n)$ be a basis for a Banach space $X$ and let $Y=(\sum\_{n=0}^\infty E\_n)\_0$ be the $c\_0$-sum of $E\_n := \text{span} \{ x\_0,\dots,x\_n\}$. Let $(y\_k)$ be the obvious basis for $Y$ induced by concatenating the given bases for $E\_n$; $n=1,2,\dots$. Then
$\text{bc}((y\_n) \ge \text{bc} (x\_n)$.


In your other post about defining a quantitative version of non bounded completeness you pointed out that there is a basis $(x\_n)$ for $c$ for which $\text{bc} (x\_n)=2$. For this basis, $E\_n := \text{span} \{ x\_0,\dots,x\_n\}$ is obviously isometrically isomorphic to $\ell\_\infty^{ n+1}$ and hence $Y:= (\sum\_{n=0}^\infty E\_n)\_0$ is isometrically isomorphic to $c\_0$.


