Counterexample of $X = M oplus N$, where $X$ is normed.
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We need to find a counter example for $X = M oplus N$ , i.e. we have $X$ given normed space and $M$ is closed subspace of $X$ , then there is no closed subspace $N$ such as $X=Moplus N$.
Obviously , $N $ can't be finite dimensional , so complement of M should be infinite dimensional and open.
I've thought to consider $C[0,1]$ but what about $M$ set? Maybe it's good to take some special functions like ${ sin(kx), cos(kx) }$?
real-analysis functional-analysis
asked Nov 24 at 9:42
openspace
3,4082822
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We need to find a counter example for $X = M oplus N$ , i.e. we have $X$ given normed space and $M$ is closed subspace of $X$ , then there is no closed subspace $N$ such as $X=Moplus N$.
Obviously , $N $ can't be finite dimensional , so complement of M should be infinite dimensional and open.
I've thought to consider $C[0,1]$ but what about $M$ set? Maybe it's good to take some special functions like ${ sin(kx), cos(kx) }$?
real-analysis functional-analysis
asked Nov 24 at 9:42
openspace
3,4082822
$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36
add a comment |
up vote
1
down vote
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up vote
1
down vote
favorite
We need to find a counter example for $X = M oplus N$ , i.e. we have $X$ given normed space and $M$ is closed subspace of $X$ , then there is no closed subspace $N$ such as $X=Moplus N$.
Obviously , $N $ can't be finite dimensional , so complement of M should be infinite dimensional and open.
I've thought to consider $C[0,1]$ but what about $M$ set? Maybe it's good to take some special functions like ${ sin(kx), cos(kx) }$?
real-analysis functional-analysis
asked Nov 24 at 9:42
openspace
3,4082822
We need to find a counter example for $X = M oplus N$ , i.e. we have $X$ given normed space and $M$ is closed subspace of $X$ , then there is no closed subspace $N$ such as $X=Moplus N$.
Obviously , $N $ can't be finite dimensional , so complement of M should be infinite dimensional and open.
I've thought to consider $C[0,1]$ but what about $M$ set? Maybe it's good to take some special functions like ${ sin(kx), cos(kx) }$?
real-analysis functional-analysis
real-analysis functional-analysis
asked Nov 24 at 9:42
openspace
3,4082822
asked Nov 24 at 9:42
openspace
3,4082822
asked Nov 24 at 9:42
openspace
3,4082822
asked Nov 24 at 9:42
openspace
3,4082822
asked Nov 24 at 9:42
openspace
3,4082822
3,4082822
$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36
add a comment |
$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36
$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36
add a comment |
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$M$ is not $Xsetminus N$. How do you conclude that $M$ is open if $N$ is f.d.?
– Kavi Rama Murthy
Nov 24 at 11:57
Possible duplicate of math.stackexchange.com/questions/108284/…
– Kavi Rama Murthy
Nov 24 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed
– openspace
Nov 24 at 11:58
I think the most common example is that $c_0(Bbb N)$ is closed without a complement in $ell^infty(Bbb N)$.
– s.harp
Nov 24 at 19:36