Boundary and completion of the metric space $c_{00}$
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
asked Nov 5 '18 at 2:53
Wesley
525313
add a comment |
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
asked Nov 5 '18 at 2:53
Wesley
525313
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
add a comment |
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
asked Nov 5 '18 at 2:53
Wesley
525313
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
asked Nov 5 '18 at 2:53
Wesley
525313
asked Nov 5 '18 at 2:53
Wesley
525313
asked Nov 5 '18 at 2:53
Wesley
525313
asked Nov 5 '18 at 2:53
Wesley
525313
asked Nov 5 '18 at 2:53
Wesley
525313
525313
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
add a comment |
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
add a comment |
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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 $(c_0,d_infty)$.
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
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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 $(c_0,d_infty)$.
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
add a comment |
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 $(c_0,d_infty)$.
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
add a comment |
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 $(c_0,d_infty)$.
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
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 $(c_0,d_infty)$.
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
answered Nov 29 '18 at 12:46
José Carlos Santos
151k22123224
151k22123224
add a comment |
add a comment |
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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