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## 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?