Can we isometrically embed $(c_{00},d_1)^*$, the completion of space of eventually null sequences of real numbers with $d_1=\sum_{n=1}^{\infty}|f(n)-g(n)|$ onto the space $l^1=\{f:\mathbb{N}\to\mathbb{R}:\sum_n|f(n)|<\infty\}$?
I think the limit functional would do the job. Is it right? This is because $c_0$ is dense in $l^1$ and $c_{00}$ is dense in $c_0$. Thus, since limit functional is an isomorphic embedding of $c_{00}$ to $c_0$ and $c_0$ is dense in $l^1$, I think the result follows. Am I right? Or should we take a cauchy sequence in $c_{00}$? Any hints? Thanks beforehand.
If you are asking if there exists an isometry from $(c_{00},d_1)$ onto $\ell^{1}$ the answer is NO. The first space is not complete and the second one is complete.
Answer for the edited version: the completion of $(c_{00},d_1)$ is exactly $\ell^{1}$.