Boundary and completion of the metric space $c_{00}$












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Identify the boundary $partial c_{00}$ in $ell^p$, for each $pin[1,infty]$. Also, for each $pin[1,infty]$, identify the completion of the metric space $(c_{00},d_p)$.



Note that $c_0$ is the set of all real sequences that converge to $0$, and note that $c_{00}:= left{x={x_n}_{n=1}^inftyin c_0,:,text{ there is an $Ninmathbb{N}$ such that $x_n=0$ for all $ngeq N$}right}$






real-analysis sequences-and-series convergence complete-spaces







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  • What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
    – michaelhowes
    Nov 5 '18 at 3:04










  • Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
    – Matematleta
    Nov 5 '18 at 3:15












  • @Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
    – Wesley
    Nov 5 '18 at 4:21


















0














Identify the boundary $partial c_{00}$ in $ell^p$, for each $pin[1,infty]$. Also, for each $pin[1,infty]$, identify the completion of the metric space $(c_{00},d_p)$.



Note that $c_0$ is the set of all real sequences that converge to $0$, and note that $c_{00}:= left{x={x_n}_{n=1}^inftyin c_0,:,text{ there is an $Ninmathbb{N}$ such that $x_n=0$ for all $ngeq N$}right}$






real-analysis sequences-and-series convergence complete-spaces







share|cite|improve this question






















  • What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
    – michaelhowes
    Nov 5 '18 at 3:04










  • Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
    – Matematleta
    Nov 5 '18 at 3:15












  • @Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
    – Wesley
    Nov 5 '18 at 4:21
















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0








0







Identify the boundary $partial c_{00}$ in $ell^p$, for each $pin[1,infty]$. Also, for each $pin[1,infty]$, identify the completion of the metric space $(c_{00},d_p)$.



Note that $c_0$ is the set of all real sequences that converge to $0$, and note that $c_{00}:= left{x={x_n}_{n=1}^inftyin c_0,:,text{ there is an $Ninmathbb{N}$ such that $x_n=0$ for all $ngeq N$}right}$






real-analysis sequences-and-series convergence complete-spaces







share|cite|improve this question













Identify the boundary $partial c_{00}$ in $ell^p$, for each $pin[1,infty]$. Also, for each $pin[1,infty]$, identify the completion of the metric space $(c_{00},d_p)$.



Note that $c_0$ is the set of all real sequences that converge to $0$, and note that $c_{00}:= left{x={x_n}_{n=1}^inftyin c_0,:,text{ there is an $Ninmathbb{N}$ such that $x_n=0$ for all $ngeq N$}right}$






real-analysis sequences-and-series convergence complete-spaces




real-analysis sequences-and-series convergence complete-spaces






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asked Nov 5 '18 at 2:53









WesleyWesley

518313




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  • What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
    – michaelhowes
    Nov 5 '18 at 3:04










  • Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
    – Matematleta
    Nov 5 '18 at 3:15












  • @Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
    – Wesley
    Nov 5 '18 at 4:21




















  • What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
    – michaelhowes
    Nov 5 '18 at 3:04










  • Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
    – Matematleta
    Nov 5 '18 at 3:15












  • @Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
    – Wesley
    Nov 5 '18 at 4:21


















What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
– michaelhowes
Nov 5 '18 at 3:04




What have you tried so far? I'd recommend starting with the cases $p=1$ and $p=infty$. Once you've done these two cases, you should see what to do for the remaining $p$.
– michaelhowes
Nov 5 '18 at 3:04












Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
– Matematleta
Nov 5 '18 at 3:15






Hint: $c_{00}$ is dense in $ell^p$ if $p<infty$ and in $c_0$ if $p=infty.$
– Matematleta
Nov 5 '18 at 3:15














@Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
– Wesley
Nov 5 '18 at 4:21






@Matematleta With regard to the question of completion, there is a theorem which states that every metric space (M,d) has a completion (M*,d*) such that M is dense in M*. Is this result relevant? So if I know that $c_{00}$ is dense in $ell^p$, do we know that the completion of $(c_{00}, d_p)$ is $(ell^p, d_p)$ when $p<infty$?
– Wesley
Nov 5 '18 at 4:21












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If $p<infty$, then $overline{c_{00}}=ell^p$ and $mathring{c_{00}}=emptyset$. Therefore, $partial c_{00}=ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(ell^p,d_p)$.



In $(ell^infty,d_infty)$, it is still true that $mathring{c_{00}}=emptyset$, but now $overline{c_{00}}=c_0$. So, $partial c_{00}=c_0$ and the completion of $(c_{00},d_infty)$ can be identified with $(c_0,d_infty)$.






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    If $p<infty$, then $overline{c_{00}}=ell^p$ and $mathring{c_{00}}=emptyset$. Therefore, $partial c_{00}=ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(ell^p,d_p)$.



    In $(ell^infty,d_infty)$, it is still true that $mathring{c_{00}}=emptyset$, but now $overline{c_{00}}=c_0$. So, $partial c_{00}=c_0$ and the completion of $(c_{00},d_infty)$ can be identified with $(c_0,d_infty)$.






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      If $p<infty$, then $overline{c_{00}}=ell^p$ and $mathring{c_{00}}=emptyset$. Therefore, $partial c_{00}=ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(ell^p,d_p)$.



      In $(ell^infty,d_infty)$, it is still true that $mathring{c_{00}}=emptyset$, but now $overline{c_{00}}=c_0$. So, $partial c_{00}=c_0$ and the completion of $(c_{00},d_infty)$ can be identified with $(c_0,d_infty)$.






      share|cite|improve this answer
























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        If $p<infty$, then $overline{c_{00}}=ell^p$ and $mathring{c_{00}}=emptyset$. Therefore, $partial c_{00}=ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(ell^p,d_p)$.



        In $(ell^infty,d_infty)$, it is still true that $mathring{c_{00}}=emptyset$, but now $overline{c_{00}}=c_0$. So, $partial c_{00}=c_0$ and the completion of $(c_{00},d_infty)$ can be identified with $(c_0,d_infty)$.






        share|cite|improve this answer












        If $p<infty$, then $overline{c_{00}}=ell^p$ and $mathring{c_{00}}=emptyset$. Therefore, $partial c_{00}=ell^p$ and the completion of $(c_{00},d_p)$ can be identified with $(ell^p,d_p)$.



        In $(ell^infty,d_infty)$, it is still true that $mathring{c_{00}}=emptyset$, but now $overline{c_{00}}=c_0$. So, $partial c_{00}=c_0$ and the completion of $(c_{00},d_infty)$ can be identified with $(c_0,d_infty)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 29 '18 at 12:46









        José Carlos SantosJosé Carlos Santos

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        152k22123225






























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