Interior of the set of "finite" sequences

[Mr Davis 97]

Homework Statement

Identify the boundary $\partial c_{00}$ in $\ell^p$, for each $p\in[1,\infty]$

Homework Equations

The interior of $S$ is $\operatorname{int}(S) = \{a\in S \mid \exists \delta > 0 \text{ such that } B_\delta (a) \subseteq S\}$.

$\partial S = \bar{S}\setminus \operatorname{int}(S)$

The Attempt at a Solution

This problem uses part of the result from the last problem I posted, but I am going to try it first since it seems a bit easier. Suppose I already know that the closure of $c_{00}$ when $p\not = \infty$ is $\ell^p$, and that the closure of $c_{00}$ when $p=\infty$ is $c_0$.

Case 1: $p\not =\infty$. $\partial c_{00} = \bar{c_{00}}\setminus \operatorname{int}(c_{00}) = \ell^p\setminus \operatorname{int}(c_{00})$.

Case 2: $p = \infty$. $\partial c_{00} = \bar{c_{00}}\setminus \operatorname{int}(c_{00}) = c_0\setminus \operatorname{int}(c_{00})$So I guess my question then is how would I go about finding the interior of $c_{00}$ in both cases?

 

[fresh_42]

$c_{00} := \left\{x=\{x_n\}_{n=1}^\infty \in \ell^p \,:\, \text{ there is an }N\in\mathbb{N} \text{ such that }x_n=0 \text{ for all }n\geq N \,\right\} \subseteq \ell^p:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\, \sum_{n\in\mathbb{N}} |x_n|^p <\infty\right\}$

$c_{00} := \left\{x=\{x_n\}_{n=1}^\infty\in \ell^\infty\,:\,\text{ there is an N\in\mathbb{N} such that x_n=0 for all n\geq N}\right\} \subseteq \ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\, \sup(\{|x_n|:n\in\mathbb{N}\})<\infty\right\}$

$c_0:= \left\{x=\{x_n\}_{n=1}^\infty\in c:\,\lim_{n\to\infty}x_n=0\right\}$

 

[Mr Davis 97]

$c_{00} := \left\{x=\{x_n\}_{n=1}^\infty \in \ell^p \,:\, \text{ there is an }N\in\mathbb{N} \text{ such that }x_n=0 \text{ for all }n\geq N \,\right\} \subseteq \ell^p:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\, \sum_{n\in\mathbb{N}} |x_n|^p <\infty\right\}$

$c_{00} := \left\{x=\{x_n\}_{n=1}^\infty\in \ell^\infty\,:\,\text{ there is an N\in\mathbb{N} such that x_n=0 for all n\geq N}\right\} \subseteq \ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\, \sup(\{|x_n|:n\in\mathbb{N}\})<\infty\right\}$

$c_0:= \left\{x=\{x_n\}_{n=1}^\infty\in c:\,\lim_{n\to\infty}x_n=0\right\}$

Could you explain a bit more?

 

[fresh_42]

Could you explain a bit more?

I simply wanted to make your thread readable. As far as I know, the notation $\ell^p$ is standard, the subsets are not. And the definitions of $c_0$ and $c_{00}$ are essential to the thread, and you have neither linked their definition nor repeated them. That's why I copied the definitions from the other thread.

I haven't thought about the topological properties of these sets. If you show your closure proofs, then they might contain a hint what the boundary, resp. the interiors are. I do not assume isolated points here, so the question will be: are those sets open?