Exercise 4.33 in Brezis functional analysis.












2














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





























    2














    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



























      2












      2








      2







      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 25 at 5:31









      Aweygan

      13.4k21441




      13.4k21441










      asked Nov 25 at 5:22









      Euduardo

      1217




      1217






















          1 Answer
          1






          active

          oldest

          votes


















          2














          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











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012468%2fexercise-4-33-in-brezis-functional-analysis%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          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
















          2














          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














          2












          2








          2






          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

          13.4k21441




          13.4k21441












          • 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


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012468%2fexercise-4-33-in-brezis-functional-analysis%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Ellipse (mathématiques)

          Quarter-circle Tiles

          Mont Emei