Consider the Banach space $\ell^1(\mathbb N)$.
The sequence $(e_n)_{n\in\mathbb N}$ feels like a kind of basis because every element $a\in\ell^1(\mathbb N)$ can be written as an absolutely convergent infinite linear combination $\sum_{n\in\mathbb N}a(n)e_n$ in a unique way.
(Here $e_n$ denotes the vector whose $n$th entry is 1 and all of whose other entries vanish.)
The same is true for the Banach space $c_0(\mathbb N)$.
Is the above property of the sequence $(e_n)_{n\in\mathbb N}$ appropriate in order to abstractly define a basis of a Banach space? Has this been considered?