Find the completion of the metric space.

Question

Let $\ell^1(\mathbb{N})=$ $\{(a_n): a_n ∈ \mathbb{R} , \sum |a_n| \lt \infty \}$, and define the distance function $d, d_* : \ell^1(\mathbb{N})\times\ell^1(\mathbb{N}) → [0, \infty)$ by

$d(a,b)=\sum |a_n - b_n|$, $d_*(a,b)= sup \,|a_n - b_n|, a, b ∈ \ell^1(\mathbb{N})$.

I'd like to find the completion of the metric space $(\ell^1(\mathbb{N}), d_*)$.

I don't know how to start...

Could you give me a hint or any help will be appreciated.

Answer 1

The completion is $c_0$, the space of sequences with limit $0$. Note that if $a_n \to 0$ then $(a_1,a_2,...,a_k,0,0,...)$ is a sequence in $\ell^{1} $ converging in the sup metric to $(a_n)$. Conversely If you can approximate $(a_n)$ under $d_{*}$ by an element of $\ell^{1}$ then $a_n \to 0$ because every sequence in $\ell^{1}$ has limit $0$.