Let $\;\;\; \big\langle V,+,\cdot, \;\; ||\cdot|| \;\; \big\rangle \;\;\;$ be a Banach space over the field $\mathbb{K}$, which is either $\mathbb{R}$ or $\mathbb{C}$.

Suppose there exists a subset $B\:$ of $V\:$ such that:

For all functions $\: f : B\to \mathbb{K} \:$, $\:$ if $\;\; \displaystyle\sum_{b\in B} \; (f(b) \cdot b) \;\;$ exists and is the zero vector, then $f\:$ is identically zero.

and

For all members $v$ of $V$, $\:$ there exists a function $\; f : B\to \mathbb{K} \;$ such that $\;\;\;\; \displaystyle\sum_{b\in B} \; (f(b) \cdot b) \;\; = \;\; v \;\;\;\;$.

Does it follow that all such subsets $B\:$ have the same cardinality?