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Interior of the set of "finite" sequences

  • #1
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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?
 

Answers and Replies

  • #2
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##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\}##
 
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  • #3
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##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?
 
  • #4
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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?
 
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