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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.

Community
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user149418
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2 Answers2

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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.

Robert Israel
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  • Without assuming that $N$ is closed, how to prove that $\pi_1$ is continuous? – user149418 Feb 18 '15 at 18:24
  • It's usually part of the definition of $\oplus$ in Banach spaces. – Robert Israel Feb 18 '15 at 18:30
  • @RobertIsrael: Can you elaborate on how to see that the projection is continuous, please? (If it were not by definition.) – freishahiri Feb 19 '15 at 22:08
  • How do you want to define $X = M \oplus N$? – Robert Israel Feb 19 '15 at 22:31
  • 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). – Rob Arthan Apr 08 '17 at 20:53
  • Hmm, you seem to be right. The definition of $M \oplus N$ in Rudin, "Functional Analysis" is indeed that way. – Robert Israel Apr 09 '17 at 04:48
  • @RobArthan: The categorical definition of a direct sum in a additive (or preadditive) category is usually (and originally!) given in terms of a pair of morphisms, $\pi_i:M_1\oplus M_2\to M_i$, called "projections", as well as a pair of morphisms, $\iota_i:M_i\to M_1\oplus M_2$, called "injections", for $i=1,2$, respectively, that satisfy the direct sum relations $\iota_1\pi_1+\iota_2\pi_2=\operatorname{id}_{M_1\oplus M_2}$ as well as $\pi_i\iota_i=\operatorname{id}_i$ for $i=1,2$. – freishahiri Dec 21 '17 at 10:58
  • For a reference see "Mac Lane, Categories for the Working Mathematician, chapter IIIV, section 2, Additive Categories". – freishahiri Dec 21 '17 at 11:09
  • @RobertIsrael: I just noticed that I answered my question in the comment, upsi.. ^^ By the way, nice answer! (+1) – freishahiri Dec 21 '17 at 14:21
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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.

Harshavardhan
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