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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?

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    Two books you should look at: 1. The classic: [Lindenstrauss-Tzafriri](http://books.google.com/books?id=v9in9nUW5bsC), 2. A nice new one: [Albiac-Kalton](http://books.google.com/books?id=FkjCulO5xg0C). Two links you should follow: 3. [Per Enflo](http://en.wikipedia.org/wiki/Per_Enflo) 4. [Approximation property](http://en.wikipedia.org/wiki/Approximation_property)2011-09-02

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As Soarer points out: Yes, it is called a Schauder basis.