A question about complement of a closed subspace of a Banach space Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Example of a closed subspace of a Banach space which is not complemented?Algebraic and topological complements in a Hilbert spaceClosed subspace of Banach spaceFind a closed subspace $M$ and a non-closed subspace $N$ of a Banach space $mathbfB$ such that $Moplus N=mathbfB$If $Moplus N=B$ is a Banach space and $M$ is closed, does that imply that $N$ is closed as well?Is there a non-reflexive Banach space with every proper closed subspace reflexive?Example of a closed subspace of a Banach space which is not complemented?Finding the topological complement of a finite dimensional subspaceIs the complement of a finite dimensional subspace always closed?Every closed separable subspace is complementedIf $E$ is not complemented in $X$, is $E oplus 0$ not complemented in $X oplus Y$?Does there exist a Banach space with no complemented closed subspaces?Closed subspace of Banach spaceBanach space with non-complemented subspaceAn exercise with finite subspace of a Banach Space
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A question about complement of a closed subspace of a Banach space
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Example of a closed subspace of a Banach space which is not complemented?Algebraic and topological complements in a Hilbert spaceClosed subspace of Banach spaceFind a closed subspace $M$ and a non-closed subspace $N$ of a Banach space $mathbfB$ such that $Moplus N=mathbfB$If $Moplus N=B$ is a Banach space and $M$ is closed, does that imply that $N$ is closed as well?Is there a non-reflexive Banach space with every proper closed subspace reflexive?Example of a closed subspace of a Banach space which is not complemented?Finding the topological complement of a finite dimensional subspaceIs the complement of a finite dimensional subspace always closed?Every closed separable subspace is complementedIf $E$ is not complemented in $X$, is $E oplus 0$ not complemented in $X oplus Y$?Does there exist a Banach space with no complemented closed subspaces?Closed subspace of Banach spaceBanach space with non-complemented subspaceAn exercise with finite subspace of a Banach Space
$begingroup$
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 $X=Moplus N$. 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.
functional-analysis banach-spaces
edited Apr 13 '17 at 12:20
Community♦
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
$endgroup$
add a comment |
$begingroup$
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 $X=Moplus N$. 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.
functional-analysis banach-spaces
edited Apr 13 '17 at 12:20
Community♦
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
$endgroup$
add a comment |
$begingroup$
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 $X=Moplus N$. 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.
functional-analysis banach-spaces
edited Apr 13 '17 at 12:20
Community♦
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
$endgroup$
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 $X=Moplus N$. 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.
functional-analysis banach-spaces
functional-analysis banach-spaces
edited Apr 13 '17 at 12:20
Community♦
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
edited Apr 13 '17 at 12:20
Community♦
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
edited Apr 13 '17 at 12:20
Community♦
1
edited Apr 13 '17 at 12:20
Community♦
1
edited Apr 13 '17 at 12:20
Community♦
1
1
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
asked Feb 18 '15 at 18:09
user149418user149418
1,596517
1,596517
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Writing $X = M oplus N$ implies that the projection $pi_1: X to M$ is continuous. Then $N = pi^-1(0)$ must be closed.
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
$endgroup$
$begingroup$
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
|
show 4 more comments
$begingroup$
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.
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Writing $X = M oplus N$ implies that the projection $pi_1: X to M$ is continuous. Then $N = pi^-1(0)$ must be closed.
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
$endgroup$
$begingroup$
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
|
show 4 more comments
$begingroup$
Writing $X = M oplus N$ implies that the projection $pi_1: X to M$ is continuous. Then $N = pi^-1(0)$ must be closed.
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
$endgroup$
$begingroup$
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
|
show 4 more comments
$begingroup$
Writing $X = M oplus N$ implies that the projection $pi_1: X to M$ is continuous. Then $N = pi^-1(0)$ must be closed.
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
$endgroup$
Writing $X = M oplus N$ implies that the projection $pi_1: X to M$ is continuous. Then $N = pi^-1(0)$ must be closed.
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
answered Feb 18 '15 at 18:17
Robert IsraelRobert Israel
332k23222481
332k23222481
$begingroup$
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
|
show 4 more comments
$begingroup$
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
$begingroup$
Without assuming that $N$ is closed, how to prove that $pi_1$ is continuous?
$endgroup$
– user149418
Feb 18 '15 at 18:24
$begingroup$
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
$begingroup$
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
$begingroup$
@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
$begingroup$
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
$begingroup$
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
|
show 4 more comments
$begingroup$
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.
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
$endgroup$
add a comment |
$begingroup$
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.
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
$endgroup$
add a comment |
$begingroup$
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.
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
$endgroup$
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.
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
answered Apr 1 at 22:09
HarshavardhanHarshavardhan
111
111
add a comment |
add a comment |
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