Let $\ell\_{1,0}(S)=(c\_{0,0}(S),\Vert\cdot\Vert\_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell\_1$ norm. Could you answer the at least one of the following questions 


1) which subspaces in $X$ are complementable?


2) which subspaces in $X$ are $C$-complementable, i.e. there exist projection with norm $\leq C$?


3) which subspaces in $X$ are $C$-complementable for all $C>1$?


4) I think that spaces $\ell\_{1,0}(S')$ with $S'\subset S$ will fit. But are there other examples?



Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$
Just some simple remarks:


1. If $P$ is projection on $X$, then $P$ extends to a projection on the completion $\ell\_(S)$ with the same norm. Therefore, if $P$ has norm one, the closure of its range in $\ell\_(S)$ is the closed span of disjointly supported functions. The only way this can happens is for the range of $P$ itself to be linearly spanned by disjointly supported functions and hence be isometrically isomorphic to $\ell\_1(T)$ for some set $T$.
2. From the first sentence of (1.) you get than any complemented subspace of $X$ has its completion isomorphic to $\ell\_1(T)$ for some set $T$.
3. Any subspace of $X$ spanned by disjointly supported functions is norm one complemented. That answers 4).
4. A subspace of $X$ is $C$-complementable for all $C>1$ iff it is $1$-complementable iff it is spanned by disjointly supported functions. Again, just pass to the closure in $\ell\_1(S)$ and use the known theory there, which I presume you know something about, else why this question?


Finally, why DID you ask this question? 


