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### Closed subspace of Banach space [duplicate]

Let S closed subspace of E, where E is Banach space. (1): Is there a decomposition of E in direct sum, E = S $\oplus$ N ? (2): If (1) holds, we have that N has linear homeomorphism with E/S. I ...
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### A question about complement of a closed subspace of a Banach space

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 ...
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### Does there exist a Banach space with no complemented closed subspaces?

I know that every Hilbert space can be decomposed as the direct sum of two non-trivial closed subspaces, eg. taking the kernel and range of any non-trivial bounded projection. But I don't know what ...
784 views

### Conditions for a kernel of a bounded operator to be complemented

I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here . Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
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### Banach space with non-complemented subspace

I see examples on Stack Exchange and elsewhere of Banach spaces with non-complemented subspaces (examples: 1, 2 [Remark 8], 3 [a remarkable example of a Banach space with no complemented closed ...
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### Closed Subspace of a Banach Space with a Non-closed Linear Complement [closed]

What is an example of a closed subspace of a Banach space whose linear complement (direct sum decomposition) is not closed?
Motivation: If $a$ and $b \ne 0$ are real numbers, then $a = b \cdot (a / b)$. Question: Let $X$ be a Banach space and $M \subset X$ a closed subspace. Then, the quotient space $X / M$ is also a ...