Does $L_1$ convergence of continuous functions imply pointwise convergence?












3












$begingroup$


Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?



I'm pretty sure the answer is no, but I'm not aware of a counterexample.






functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence







share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?



    I'm pretty sure the answer is no, but I'm not aware of a counterexample.






    functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?



      I'm pretty sure the answer is no, but I'm not aware of a counterexample.






      functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence







      share|cite|improve this question









      $endgroup$




      Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?



      I'm pretty sure the answer is no, but I'm not aware of a counterexample.






      functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence




      functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 24 at 2:52









      Keshav SrinivasanKeshav Srinivasan

      2,33421445




      2,33421445






















          2 Answers
          2






          active

          oldest

          votes


















          6












          $begingroup$

          Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            The "dancing indicators" counterexample:
            $$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$



            Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              well, those are not continuous
              $endgroup$
              – zhw.
              Jan 24 at 4:20






            • 1




              $begingroup$
              If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
              $endgroup$
              – eepperly16
              Jan 24 at 4:28











            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%2f3085389%2fdoes-l-1-convergence-of-continuous-functions-imply-pointwise-convergence%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            6












            $begingroup$

            Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.






                share|cite|improve this answer









                $endgroup$



                Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 24 at 4:23









                zhw.zhw.

                74.2k43175




                74.2k43175























                    3












                    $begingroup$

                    The "dancing indicators" counterexample:
                    $$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$



                    Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.






                    share|cite|improve this answer











                    $endgroup$









                    • 1




                      $begingroup$
                      well, those are not continuous
                      $endgroup$
                      – zhw.
                      Jan 24 at 4:20






                    • 1




                      $begingroup$
                      If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                      $endgroup$
                      – eepperly16
                      Jan 24 at 4:28
















                    3












                    $begingroup$

                    The "dancing indicators" counterexample:
                    $$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$



                    Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.






                    share|cite|improve this answer











                    $endgroup$









                    • 1




                      $begingroup$
                      well, those are not continuous
                      $endgroup$
                      – zhw.
                      Jan 24 at 4:20






                    • 1




                      $begingroup$
                      If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                      $endgroup$
                      – eepperly16
                      Jan 24 at 4:28














                    3












                    3








                    3





                    $begingroup$

                    The "dancing indicators" counterexample:
                    $$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$



                    Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.






                    share|cite|improve this answer











                    $endgroup$



                    The "dancing indicators" counterexample:
                    $$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$



                    Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Jan 24 at 7:24

























                    answered Jan 24 at 3:05









                    angryavianangryavian

                    42.1k23381




                    42.1k23381








                    • 1




                      $begingroup$
                      well, those are not continuous
                      $endgroup$
                      – zhw.
                      Jan 24 at 4:20






                    • 1




                      $begingroup$
                      If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                      $endgroup$
                      – eepperly16
                      Jan 24 at 4:28














                    • 1




                      $begingroup$
                      well, those are not continuous
                      $endgroup$
                      – zhw.
                      Jan 24 at 4:20






                    • 1




                      $begingroup$
                      If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                      $endgroup$
                      – eepperly16
                      Jan 24 at 4:28








                    1




                    1




                    $begingroup$
                    well, those are not continuous
                    $endgroup$
                    – zhw.
                    Jan 24 at 4:20




                    $begingroup$
                    well, those are not continuous
                    $endgroup$
                    – zhw.
                    Jan 24 at 4:20




                    1




                    1




                    $begingroup$
                    If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                    $endgroup$
                    – eepperly16
                    Jan 24 at 4:28




                    $begingroup$
                    If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
                    $endgroup$
                    – eepperly16
                    Jan 24 at 4:28


















                    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.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3085389%2fdoes-l-1-convergence-of-continuous-functions-imply-pointwise-convergence%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

                    Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

                    Image
                    28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
                    Read more

                    Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

                    Image
                    up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
                    Read more

                    A Topological Invariant for $pi_3(U(n))$

                    Image
                    3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
                    Read more