Exercise 4.33 in Brezis functional analysis.











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Fix a function $phi in C_c(mathbb{R}), phinotequiv0,$ and consider the family of functions



$mathcal{F} = {phi_n:ninmathbb N},$ where $phi_n(x) = phi(x+n), xinmathbb{R}.$



The problem is to prove that $mathcal{F}$ does not have compact closure in $L^p(mathbb{R}) (1le p <infty),$ but there is a theorem that the closure $mathcal{F}|_{Omega}$ in $L^p(Omega)$ is compact for any finite-measure subset $Omegainmathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $mathbb{R}.$



I think I need to induce contradiction assuming $mathcal{F}$ has compact closure in $L^p(mathbb{R}),$ but I cannot come up with the next step.



Any hint or idea would be appreciated. Thanks in advance.










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    up vote
    2
    down vote

    favorite












    Fix a function $phi in C_c(mathbb{R}), phinotequiv0,$ and consider the family of functions



    $mathcal{F} = {phi_n:ninmathbb N},$ where $phi_n(x) = phi(x+n), xinmathbb{R}.$



    The problem is to prove that $mathcal{F}$ does not have compact closure in $L^p(mathbb{R}) (1le p <infty),$ but there is a theorem that the closure $mathcal{F}|_{Omega}$ in $L^p(Omega)$ is compact for any finite-measure subset $Omegainmathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $mathbb{R}.$



    I think I need to induce contradiction assuming $mathcal{F}$ has compact closure in $L^p(mathbb{R}),$ but I cannot come up with the next step.



    Any hint or idea would be appreciated. Thanks in advance.










    share|cite|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Fix a function $phi in C_c(mathbb{R}), phinotequiv0,$ and consider the family of functions



      $mathcal{F} = {phi_n:ninmathbb N},$ where $phi_n(x) = phi(x+n), xinmathbb{R}.$



      The problem is to prove that $mathcal{F}$ does not have compact closure in $L^p(mathbb{R}) (1le p <infty),$ but there is a theorem that the closure $mathcal{F}|_{Omega}$ in $L^p(Omega)$ is compact for any finite-measure subset $Omegainmathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $mathbb{R}.$



      I think I need to induce contradiction assuming $mathcal{F}$ has compact closure in $L^p(mathbb{R}),$ but I cannot come up with the next step.



      Any hint or idea would be appreciated. Thanks in advance.










      share|cite|improve this question















      Fix a function $phi in C_c(mathbb{R}), phinotequiv0,$ and consider the family of functions



      $mathcal{F} = {phi_n:ninmathbb N},$ where $phi_n(x) = phi(x+n), xinmathbb{R}.$



      The problem is to prove that $mathcal{F}$ does not have compact closure in $L^p(mathbb{R}) (1le p <infty),$ but there is a theorem that the closure $mathcal{F}|_{Omega}$ in $L^p(Omega)$ is compact for any finite-measure subset $Omegainmathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $mathbb{R}.$



      I think I need to induce contradiction assuming $mathcal{F}$ has compact closure in $L^p(mathbb{R}),$ but I cannot come up with the next step.



      Any hint or idea would be appreciated. Thanks in advance.







      functional-analysis lp-spaces






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      edited Nov 25 at 5:31









      Aweygan

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      asked Nov 25 at 5:22









      Euduardo

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          Hint: We have $operatorname{supp}phisubset [-N,N]$ for some $Ninmathbb N$. Can the sequence ${phi_{nN}}_{ninmathbb N}$ have a convergent subsequence?






          share|cite|improve this answer





















          • Your answer makes me feel I am stupid... Thank you Aweygan!
            – Euduardo
            Nov 25 at 5:33






          • 1




            You're welcome, but don't feel stupid!
            – Aweygan
            Nov 25 at 5:36











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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          Hint: We have $operatorname{supp}phisubset [-N,N]$ for some $Ninmathbb N$. Can the sequence ${phi_{nN}}_{ninmathbb N}$ have a convergent subsequence?






          share|cite|improve this answer





















          • Your answer makes me feel I am stupid... Thank you Aweygan!
            – Euduardo
            Nov 25 at 5:33






          • 1




            You're welcome, but don't feel stupid!
            – Aweygan
            Nov 25 at 5:36















          up vote
          2
          down vote



          accepted










          Hint: We have $operatorname{supp}phisubset [-N,N]$ for some $Ninmathbb N$. Can the sequence ${phi_{nN}}_{ninmathbb N}$ have a convergent subsequence?






          share|cite|improve this answer





















          • Your answer makes me feel I am stupid... Thank you Aweygan!
            – Euduardo
            Nov 25 at 5:33






          • 1




            You're welcome, but don't feel stupid!
            – Aweygan
            Nov 25 at 5:36













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          Hint: We have $operatorname{supp}phisubset [-N,N]$ for some $Ninmathbb N$. Can the sequence ${phi_{nN}}_{ninmathbb N}$ have a convergent subsequence?






          share|cite|improve this answer












          Hint: We have $operatorname{supp}phisubset [-N,N]$ for some $Ninmathbb N$. Can the sequence ${phi_{nN}}_{ninmathbb N}$ have a convergent subsequence?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 25 at 5:31









          Aweygan

          13.1k21441




          13.1k21441












          • Your answer makes me feel I am stupid... Thank you Aweygan!
            – Euduardo
            Nov 25 at 5:33






          • 1




            You're welcome, but don't feel stupid!
            – Aweygan
            Nov 25 at 5:36


















          • Your answer makes me feel I am stupid... Thank you Aweygan!
            – Euduardo
            Nov 25 at 5:33






          • 1




            You're welcome, but don't feel stupid!
            – Aweygan
            Nov 25 at 5:36
















          Your answer makes me feel I am stupid... Thank you Aweygan!
          – Euduardo
          Nov 25 at 5:33




          Your answer makes me feel I am stupid... Thank you Aweygan!
          – Euduardo
          Nov 25 at 5:33




          1




          1




          You're welcome, but don't feel stupid!
          – Aweygan
          Nov 25 at 5:36




          You're welcome, but don't feel stupid!
          – Aweygan
          Nov 25 at 5:36


















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