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?