Equivalent operator norm
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Let $E$ be a Banach space and let $L:E to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${left| {Lx} right|_E}$ and ${left| {x} right|_E}$. More precisely,what is the condition on $L$ to obtain a such equivalence. Thank you.
real-analysis functional-analysis operator-theory normed-spaces
asked Nov 22 at 19:28
Gustave
692211
add a comment |
up vote
1
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Let $E$ be a Banach space and let $L:E to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${left| {Lx} right|_E}$ and ${left| {x} right|_E}$. More precisely,what is the condition on $L$ to obtain a such equivalence. Thank you.
real-analysis functional-analysis operator-theory normed-spaces
asked Nov 22 at 19:28
Gustave
692211
2
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $E$ be a Banach space and let $L:E to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${left| {Lx} right|_E}$ and ${left| {x} right|_E}$. More precisely,what is the condition on $L$ to obtain a such equivalence. Thank you.
real-analysis functional-analysis operator-theory normed-spaces
asked Nov 22 at 19:28
Gustave
692211
Let $E$ be a Banach space and let $L:E to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${left| {Lx} right|_E}$ and ${left| {x} right|_E}$. More precisely,what is the condition on $L$ to obtain a such equivalence. Thank you.
real-analysis functional-analysis operator-theory normed-spaces
real-analysis functional-analysis operator-theory normed-spaces
asked Nov 22 at 19:28
Gustave
692211
asked Nov 22 at 19:28
Gustave
692211
asked Nov 22 at 19:28
Gustave
692211
asked Nov 22 at 19:28
Gustave
692211
asked Nov 22 at 19:28
Gustave
692211
692211
2
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54
add a comment |
2
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54
2
2
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54
add a comment |
1 Answer
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If I understood you correctly, you want to find a characterization of bounded linear maps $L:Eto E$ such that there exist constants $c, C > 0$ such that
$$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert. $$
Notice that the right inequality is trivial, since $L$ is bounded.
As for the left inequality, it is a nice exercise to show that the following are equivalent:
$L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;
$L$ is injective and its range is closed;
The transpose $L':E'to E'$ of $L$ is surjective.
answered Nov 22 at 19:59
MisterRiemann
5,7131624
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
add a comment |
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1 Answer
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If I understood you correctly, you want to find a characterization of bounded linear maps $L:Eto E$ such that there exist constants $c, C > 0$ such that
$$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert. $$
Notice that the right inequality is trivial, since $L$ is bounded.
As for the left inequality, it is a nice exercise to show that the following are equivalent:
$L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;
$L$ is injective and its range is closed;
The transpose $L':E'to E'$ of $L$ is surjective.
answered Nov 22 at 19:59
MisterRiemann
5,7131624
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
add a comment |
up vote
1
down vote
If I understood you correctly, you want to find a characterization of bounded linear maps $L:Eto E$ such that there exist constants $c, C > 0$ such that
$$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert. $$
Notice that the right inequality is trivial, since $L$ is bounded.
As for the left inequality, it is a nice exercise to show that the following are equivalent:
$L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;
$L$ is injective and its range is closed;
The transpose $L':E'to E'$ of $L$ is surjective.
answered Nov 22 at 19:59
MisterRiemann
5,7131624
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
add a comment |
up vote
1
down vote
up vote
1
down vote
If I understood you correctly, you want to find a characterization of bounded linear maps $L:Eto E$ such that there exist constants $c, C > 0$ such that
$$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert. $$
Notice that the right inequality is trivial, since $L$ is bounded.
As for the left inequality, it is a nice exercise to show that the following are equivalent:
$L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;
$L$ is injective and its range is closed;
The transpose $L':E'to E'$ of $L$ is surjective.
answered Nov 22 at 19:59
MisterRiemann
5,7131624
If I understood you correctly, you want to find a characterization of bounded linear maps $L:Eto E$ such that there exist constants $c, C > 0$ such that
$$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert. $$
Notice that the right inequality is trivial, since $L$ is bounded.
As for the left inequality, it is a nice exercise to show that the following are equivalent:
$L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;
$L$ is injective and its range is closed;
The transpose $L':E'to E'$ of $L$ is surjective.
answered Nov 22 at 19:59
MisterRiemann
5,7131624
answered Nov 22 at 19:59
MisterRiemann
5,7131624
answered Nov 22 at 19:59
MisterRiemann
5,7131624
answered Nov 22 at 19:59
MisterRiemann
5,7131624
5,7131624
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
add a comment |
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
Thanks.I have solved this exercice in the past and I forgot.
– Gustave
Nov 22 at 20:13
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Glad I could help!
– MisterRiemann
Nov 22 at 20:14
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
@Gustave Please mark it as an answer if you're satisfied with it, to let the other users know that you received the help you needed.
– MisterRiemann
Nov 22 at 20:33
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
Thank you for the answer. I'm so sarisfied.
– Gustave
Nov 22 at 20:45
add a comment |
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2
What exactly do you mean by equivalence? Do you want equality? Or perhaps something like $$ cVert x Vert leq Vert Lx Vert leq C Vert x Vert? $$
– MisterRiemann
Nov 22 at 19:33
@MisterRiemann yes, I talked on the usual equivalence.
– Gustave
Nov 22 at 19:52
@jameswatt non. The question is as it is written.
– Gustave
Nov 22 at 19:54