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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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See Schauder basis: – Soarer Sep 4 '10 at 15:35
@KennyT, Soarer: Thank you for your help! – Rasmus Sep 4 '10 at 15:41
As pointed out by Soarer Schauder basis is what you want here, for completeness of the wiki question in the title I also would recommend the Hamel basis,, which is sometimes very useful when constructing counter examples. – AD. Sep 5 '10 at 5:06
Two books you should look at: 1. The classic: Lindenstrauss-Tzafriri, 2. A nice new one: Albiac-Kalton. Two links you should follow: 3. Per Enflo 4. Approximation property – t.b. Sep 2 '11 at 2:48
up vote 5 down vote accepted

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

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