Linked Questions

1
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0answers
265 views

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 ...
2
votes
2answers
1k views

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 ...
8
votes
1answer
525 views

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 ...
6
votes
1answer
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 $M$ ...
0
votes
2answers
568 views

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 ...
2
votes
1answer
306 views

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?
3
votes
2answers
97 views

Banach space is product of quotient space

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 ...
2
votes
0answers
360 views

nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any $x\...
2
votes
1answer
165 views

Simple example that density of the subspace cannot be omitted from the Bounded Extension from Dense Subspace Theorem.

Bounded Extension from Dense Subspace Theorem. Suppose that $Μ$ is a dense subspace of a normed space $X$, that $Y$ is a Banach space, and that $T_0: Μ \to Y$ is a bounded linear operator. Then there ...
3
votes
1answer
85 views

Recovering convergence in a Banach space from convergence in its quotient space

Suppose $X$ is Banach and $N$ its closed subspace, then $X/N$ is in turn Banach. Now suppose we have a convergent sequence $\bar{x_n}\to \bar{x_0}$ in $X/N$, is it then possible to construct a ...
0
votes
1answer
43 views

If $x_0 \in X$ and not in the closure of $M$, then there exists $T \in X^{\ast}$ such that $Tx_0 \neq 0$ and $T_{|M} = 0$

Let $X$ be a normed linear space, $M$ a linear subspace of $X$. The following proposition If $x_0 \in X$ and not in the closure of $M$, then there exists $T \in X^{\ast}$ such that $Tx_0 \neq 0$ ...
1
vote
0answers
21 views

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

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 ,...