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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 $\displaystyle\sum_{b \sum_{b \in B} c(b) \cdot b \: = 0$, then $c$ is identically zero.
• Spanning Set: For all vectors $v$ in $V$, there exists a function $c$ in $\mathbb{R}^B$ such that $\displaystyle\sum_{b \sum_{b \in B} c(b) \cdot b \: = v$.

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

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# Basis for L_infty(R)

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 $\displaystyle\sum_{b \in B} c(b) \cdot b \: = 0$, then $c$ is identically zero.

Spanning Set: For all vectors $v$ in $V$, there exists a function $c$ in $\mathbb{R}^B$ such that $\displaystyle\sum_{b \in B} c(b) \cdot b \: = v$

.

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