Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that


* Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum\_{b \in B} c(b) \cdot b = 0$, then $c$ is identically zero.
* Spanning Set: For all vectors $v$ in $V$, there exists a function $c$ in $\mathbb{R}^B$ such that $\sum\_{b \in B} c(b) \cdot b = v$.


If so, is an explicit such $B$ known?



Basis for L_infty(R)
The space $\ell^\infty\_R$ does not have even an M-basis; i.e., a biorthogonal set $(x\_t,x\_t^\*)$ such that the span of the $x\_t$ is dense and the $x\_t^\*$ are total (Lindenstrauss, late 1960s IIRC), so it has nothing like a Schauder basis. Later I proved [PAMS 26. no. 3 467-468 (1970)] that $\ell^\infty$ also does not have an M-basis. However, each of these spaces does have a biorthogonal set $(x\_t,x\_t^\*)$ such that the span of the $x\_t$ is dense. This is in my paper with W.J. Davis [Studia Math. 45 173-179 (1973)].


