In the following I will write $\mathbb{R}^{\infty}$ and $c_{00}$ to mean the following sets:
$\mathbb{R}^{\infty} = \left\{ \ x = ( x^{(1)}, x^{(2)}, x^{(3)}, \ldots, x^{(k)}, \ldots ) \mid x^{(k)} \in \mathbb{R} \text{ for all } k \in \mathbb{N} \ \right\}$
$c_{00} = \left\{ \ x = ( x^{(1)}, x^{(2)}, x^{(3)}, \dots, x^{(k)}, \dots ) \in \mathbb{R}^{\infty} \ | \ \exists K \in \mathbb{N} \mathrm{\ such\ that\ }x^{(k)}=0\mathrm{\ for\ all\ }k>K \ \right\}$
The norm I am considering is:
$\|\cdot\|_{1}$ defined by $||x\|_1 := \sum\limits_{k=1}^\infty | x^{(k)} |$
it is quite clear that $(c_{00}, \|\cdot\|_1)$ is not complete.
QUESTION: What is the completion of the vector space $(c_{00}, \|\cdot\|_1)$?
Note; When I say completion, I mean a Banach space $(X, \|\cdot\|_X)$ (aka. complete normed vector space), where:
(a) $c_{00} \subseteq X \subseteq \mathbb{R}^\infty$
(b) $\|\cdot\|_X$ extends $\|\cdot\|_1$ such that $\|x\|_X = \|x\|_1$ for all $x \in c_{00}$
(c) $c_{00}$ is dense in $X$ (IE: $\operatorname{cl}(c_{00}) = X$ in $(X, \|\cdot\|_X)$)
$\ $
MY HUNCH: I think that the completion of $(c_{00}, \|\cdot\|_1)$ is the vector space $(\ell_1, \|\cdot\|_1)$ where $\ell_1$ is the following set:
$$\ell_1 = \left\{ \ x = ( x^{(1)}, x^{(2)}, x^{(3)}, \ldots, x^{(k)}, \ldots ) \in \mathbb{R}^{\infty} \mid \|x\|_1<\infty \text{ and } \lim\limits_{k \to \infty}\{x^{(k)}\} = 0 \ \right\}$$
I am really not sure how to go about proving this however. And there is a very good chance that this is not even correct. In addition to this, I am not even sure that $(\ell_{1}, ||\cdot||_{1})$ is complete.
Can someone point me in the right direction?
Your hunch is correct. The fact that the $l^1$ is complete is a general truth about the absolutely integrable functions on a measure space (the measure, in this case, being cardinality as applied to subsets of $\mathbb N$). And the fact that $c_{00}$ is dense can be proven by approximating every absolutely summable series with its truncations (partial sums).