How to show that if $X=M oplus N$ is a Banach space then $exists c > 0$ $forall min M , forall nin N:...












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Let $X=M oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$.
How to prove that $exists c > 0$ constant such that $forall m in M$ and $forall n in N$ $$ leftlVert mrightrVert + leftlVert nrightrVert leq c leftlVert m + nrightrVert$$?






functional-analysis banach-spaces direct-sum







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closed as off-topic by amWhy, Saad, jgon, Chinnapparaj R, user10354138 Nov 26 at 8:13


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    Let $X=M oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$.
    How to prove that $exists c > 0$ constant such that $forall m in M$ and $forall n in N$ $$ leftlVert mrightrVert + leftlVert nrightrVert leq c leftlVert m + nrightrVert$$?






    functional-analysis banach-spaces direct-sum







    share|cite|improve this question













    closed as off-topic by amWhy, Saad, jgon, Chinnapparaj R, user10354138 Nov 26 at 8:13


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Saad, jgon, Chinnapparaj R, user10354138

    If this question can be reworded to fit the rules in the help center, please edit the question.
















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      Let $X=M oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$.
      How to prove that $exists c > 0$ constant such that $forall m in M$ and $forall n in N$ $$ leftlVert mrightrVert + leftlVert nrightrVert leq c leftlVert m + nrightrVert$$?






      functional-analysis banach-spaces direct-sum







      share|cite|improve this question













      Let $X=M oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$.
      How to prove that $exists c > 0$ constant such that $forall m in M$ and $forall n in N$ $$ leftlVert mrightrVert + leftlVert nrightrVert leq c leftlVert m + nrightrVert$$?






      functional-analysis banach-spaces direct-sum




      functional-analysis banach-spaces direct-sum






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      share|cite|improve this question











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      asked Nov 25 at 19:58









      kaxuzibum

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      closed as off-topic by amWhy, Saad, jgon, Chinnapparaj R, user10354138 Nov 26 at 8:13


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Saad, jgon, Chinnapparaj R, user10354138

      If this question can be reworded to fit the rules in the help center, please edit the question.




      closed as off-topic by amWhy, Saad, jgon, Chinnapparaj R, user10354138 Nov 26 at 8:13


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Saad, jgon, Chinnapparaj R, user10354138

      If this question can be reworded to fit the rules in the help center, please edit the question.






















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          Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $|m| leq c_1|m + n|$ and $|n| leq c_2|m+n|$.






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            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2














            Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $|m| leq c_1|m + n|$ and $|n| leq c_2|m+n|$.






            share|cite|improve this answer


























              2














              Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $|m| leq c_1|m + n|$ and $|n| leq c_2|m+n|$.






              share|cite|improve this answer
























                2












                2








                2






                Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $|m| leq c_1|m + n|$ and $|n| leq c_2|m+n|$.






                share|cite|improve this answer












                Try showing boundedness of the projection operators $P(m+n) = m$ and $Q(m+n) = n$ by the use of, for instance, closed graph theorem. This will show that $|m| leq c_1|m + n|$ and $|n| leq c_2|m+n|$.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered Nov 25 at 20:27









                Bartosz Malman

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                8011620















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