Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$

Question

The question is:

identify $\operatorname{co} (\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$ where $1\leq p\leq \infty$.

Notation:

$c_0$ is the collection of all sequences of scalars that converges to 0,

$\ell^p$ is the collection of all sequences $(a_j)$ of scalars for which $\sum_1^\infty |a_j|^p< \infty$ and

$\ell^\infty$ is the collection of all bounded sequences.

My Attempt:

since for any subset $A$ of a vector space, $\operatorname{co}(A)$ is the collection of all convex combination of elements of $A$ so I think $\operatorname{co}(\left\{e_n:n\in\mathbb N\}=\{(a_j)\in c_{00}: a_j\geq 0, \sum a_j=1\right\}$ in $c_0$ and $\ell^p$ spaces where

$c_{00}$ is the collection of all sequences that have only finitely many nonzero terms.

And I know $\overline{\operatorname{co}}(A)=\overline{\operatorname{co}(A)}$ in norm spaces but I don't know how to find $\overline{\operatorname{co}\{e_n: n\in \mathbb N\}}$.

Answer 1

The answer is $\{\{a_j\}:a_j \geq 0$ and $\sum a_j =1\}$. This set is contained in $c_0$ and every $l^{p}$ and sequences in this set can be approximated by similar elements of $c_{00}$.