What is the completion of the normed space? What is the completion of the normed space $C_{0,0}, ||~||_p $ ?
$(1 \leq p < \infty)$ remember that if $\overline{x} = (Xn)^\infty_{n=1}$, then $||\overline{X}||_p = (\sum_{n=1}^{\infty}|Xn|^p)^{1/p}$
We have $C_{0,0}$ is the set of sequences with entries almost all zero except for a finite quantity. Then, we have the completion of $ C_{0,0}, ||~||_p $, is the space  $lp, ||~||_p$.
I think I have to prove that $ C_{0,0} \subset lp , \forall p$ tq $(1 \leq p < \infty)$ and $ C_{0,0}$ is  dense in $lp$
I don't know how can I follow this prove, could you help me, please?
 A: Your sequence space is more commonly denoted by $c_{00} = \{(x_n) \in \Bbb C^{\Bbb N}\mid \{n: x_n \neq 0\} \text{ is finite }\}$ (or take real sequences instead of complex ones, whatever your text does).
First off, it's clear that all such sequences are in $\ell^p$ for any $p\ge 1$ because we only have a finite sum to compute for the norm. So the norm is always well-defined; in short an "eventually $0$-sequence" is always $p$-summable. for any $p$.
The density is also clear because we can, for a given $x:=(x_n)_n \in \ell^p$, define a sequence $(x_1, 0, 0, \ldots), (x_1,x_2, 0,0,\ldots), (x_1,x_2,x_3,0,0,\ldots)$ if vectors in $c_{00}$ that converges to $x$ in the $\|\cdot\|_p$-norm (if $\varepsilon>0$, there is some $N\in \Bbb N$ such that $\sum_{n=N}^\infty |x_n|^p < \varepsilon$ etc.), so that $\overline{c_{00}} = \ell^p$ for every $1 \le p < \infty$.
The completion of $c_{00}$ in $\ell^\infty$ is the space $c_0$ of seqences that converge to $0$ in the sup-norm. The arguments are quite similar.
