Counterexample of $X = M \oplus N$, where $X$ is normed.

Asked Nov 24 '18 at 09:42

Active Nov 24 '18 at 09:42

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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 $X=M\oplus N$.

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 $\{ \sin(kx), \cos(kx) \}$?

asked Nov 24 '18 at 09:42

openspace

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$M$ is not $X\setminus N$. How do you conclude that $M$ is open if $N$ is f.d.? – Kavi Rama Murthy Nov 24 '18 at 11:57
Possible duplicate of https://math.stackexchange.com/questions/108284/example-of-a-closed-subspace-of-a-banach-space-which-is-not-complemented?rq=1 – Kavi Rama Murthy Nov 24 '18 at 11:58
@KaviRamaMurthy my bad, I guessed that N should be open. It may be not open nor closed – openspace Nov 24 '18 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 '18 at 19:36

Counterexample of $X = M \oplus N$, where $X$ is normed.

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