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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 know to prove (2), but I don't know if (1) is true.
a) In case that (1) is false, if S has finite dimension, it can be true (1)?
b) In case that (1) is false. If E/S has finite dimension, where E/S has linear homeomorphism with Y, is there a decomposition of E in direct sum, E = Y $\oplus$ S ?
(In my question, I don´t ask for complemented, i.e., my N is not necessarily closed.)
For (1), it has a negative answer. Suppose it's true, so for all S closed subspace of X, Banach space, we have that exists N subspace such that E = S $\oplus$ N. But, by this argument A question about complement of a closed subspace of a Banach space, N is closed. So, every closed subspace is complemented, it's a contradiction with this Example of a closed subspace of a Banach space which is not complemented? .
For (a), I found a positive answer, here: Does there exist a Banach space with no complemented closed subspaces? , we have that "Finite-dimensional subspaces are always complemented; this follows from the Hahn–Banach theorem.".
But (b) remains open. I saw this If the quotient of a subspace of a banach space is finite, is it a closed subspace? , however it's a different question.