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Let $X$ be a Banach space and $M$ be a closed subspace of $X$. Suppose that there exists a subspace $N$ of $X$ such that $X=M\oplus N$. Does it imply that $N$ is closed ?

I know that not every closed subspace of a Banach space is complemented (see here). But my question is slightly different from that question. I think the answer is no. But I do not able to construct a counter example.

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Writing $X = M \oplus N$ implies that the projection $\pi_1: X \to M$ is continuous. Then $N = \pi^{-1}(\{0\})$ must be closed.

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  • $\begingroup$ Without assuming that $N$ is closed, how to prove that $\pi_1$ is continuous? $\endgroup$ – user149418 Feb 18 '15 at 18:24
  • $\begingroup$ It's usually part of the definition of $\oplus$ in Banach spaces. $\endgroup$ – Robert Israel Feb 18 '15 at 18:30
  • $\begingroup$ @RobertIsrael: Can you elaborate on how to see that the projection is continuous, please? (If it were not by definition.) $\endgroup$ – C-Star-W-Star Feb 19 '15 at 22:08
  • $\begingroup$ How do you want to define $X = M \oplus N$? $\endgroup$ – Robert Israel Feb 19 '15 at 22:31
  • $\begingroup$ Apologies for being pedantic 2 years on, but the general categorical definition of an (external) direct sum $M \oplus N$ of $M$ and $N$ is given in terms of injections from $M$ and $N$ to $M \oplus N$ and not in terms of projections from $M \oplus N$ to $M$ and $N$ (these arise in the definition of products). The definitions of an internal direct sum for Banach spaces that I know of don't mention projections (why would they?), but just say that it is a vector space internal direct sum whose summands are sub-Banach spaces (i.e., closed). $\endgroup$ – Rob Arthan Apr 8 '17 at 20:53
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I think the question you are trying to get at is about the relation between algebraic complements and topological complements ie if

1) M and N are complemented algebraically (complements defined without any topology involved).

2) M is closed.

Does this mean N is closed?

The answer is no, See this answer on the same site for a counterexample.

See this survey for more relations between algebraic and topological complements. In the Banach space setting, two closed subspaces are algebraic complemented if and only if they are topologically complemented.

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