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







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    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















up vote
1
down vote

favorite
1












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







share|cite|improve this question


















  • 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













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











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share|cite|improve this question










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














  • 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










1 Answer
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1
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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:




  1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;


  2. $L$ is injective and its range is closed;


  3. The transpose $L':E'to E'$ of $L$ is surjective.







share|cite|improve this answer





















  • 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











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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:




  1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;


  2. $L$ is injective and its range is closed;


  3. The transpose $L':E'to E'$ of $L$ is surjective.







share|cite|improve this answer





















  • 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















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:




  1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;


  2. $L$ is injective and its range is closed;


  3. The transpose $L':E'to E'$ of $L$ is surjective.







share|cite|improve this answer





















  • 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













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:




  1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;


  2. $L$ is injective and its range is closed;


  3. The transpose $L':E'to E'$ of $L$ is surjective.







share|cite|improve this answer












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:




  1. $L$ is bounded below, i.e. there exists a constant $c>0$ such that $c Vert xVert leq Vert LxVert$;


  2. $L$ is injective and its range is closed;


  3. The transpose $L':E'to E'$ of $L$ is surjective.








share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • 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


















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