Is the Set of Continuous Functions that are the Sum of Even and Odd Functions Meager?












6














Consider $X = mathcal{C}([−1,1])$ with the usual norm $|f|_{infty} = sup_{tin [−1,1]}|f(t)|.$



Define
$$mathcal{A}_{+}={ f in X : f(t)=f(−t) space forall tin [−1,1]},$$
$$mathcal{A}_{−}={ f in X : f(t)=−f(−t) space forall t in [−1,1]}. $$



Is $mathcal{A}_{+} +mathcal{A}_{−} = {f +g : f in mathcal{A}_{+},g in mathcal{A}_{−}}$ meager?




I know this set is dense by the Stone-Weierstrass Theorem. However, that doesn't really help. I also know that if the set is closed, then it is meager, but I have difficulties deciding whether it is closed or not. I know the exponential function is a limit of a sequence of a sum of even and odd functions, however one could define it to be that, in which case it doesn't help.




Any hints on how to get going on this problem, and on whether the set $mathcal{A}_{+}+{A}_{-} $ is closed or not? Thank you in advance.






real-analysis general-topology functional-analysis metric-spaces baire-category







share|cite|improve this question



























    6














    Consider $X = mathcal{C}([−1,1])$ with the usual norm $|f|_{infty} = sup_{tin [−1,1]}|f(t)|.$



    Define
    $$mathcal{A}_{+}={ f in X : f(t)=f(−t) space forall tin [−1,1]},$$
    $$mathcal{A}_{−}={ f in X : f(t)=−f(−t) space forall t in [−1,1]}. $$



    Is $mathcal{A}_{+} +mathcal{A}_{−} = {f +g : f in mathcal{A}_{+},g in mathcal{A}_{−}}$ meager?




    I know this set is dense by the Stone-Weierstrass Theorem. However, that doesn't really help. I also know that if the set is closed, then it is meager, but I have difficulties deciding whether it is closed or not. I know the exponential function is a limit of a sequence of a sum of even and odd functions, however one could define it to be that, in which case it doesn't help.




    Any hints on how to get going on this problem, and on whether the set $mathcal{A}_{+}+{A}_{-} $ is closed or not? Thank you in advance.






    real-analysis general-topology functional-analysis metric-spaces baire-category







    share|cite|improve this question

























      6












      6








      6


      2





      Consider $X = mathcal{C}([−1,1])$ with the usual norm $|f|_{infty} = sup_{tin [−1,1]}|f(t)|.$



      Define
      $$mathcal{A}_{+}={ f in X : f(t)=f(−t) space forall tin [−1,1]},$$
      $$mathcal{A}_{−}={ f in X : f(t)=−f(−t) space forall t in [−1,1]}. $$



      Is $mathcal{A}_{+} +mathcal{A}_{−} = {f +g : f in mathcal{A}_{+},g in mathcal{A}_{−}}$ meager?




      I know this set is dense by the Stone-Weierstrass Theorem. However, that doesn't really help. I also know that if the set is closed, then it is meager, but I have difficulties deciding whether it is closed or not. I know the exponential function is a limit of a sequence of a sum of even and odd functions, however one could define it to be that, in which case it doesn't help.




      Any hints on how to get going on this problem, and on whether the set $mathcal{A}_{+}+{A}_{-} $ is closed or not? Thank you in advance.






      real-analysis general-topology functional-analysis metric-spaces baire-category







      share|cite|improve this question













      Consider $X = mathcal{C}([−1,1])$ with the usual norm $|f|_{infty} = sup_{tin [−1,1]}|f(t)|.$



      Define
      $$mathcal{A}_{+}={ f in X : f(t)=f(−t) space forall tin [−1,1]},$$
      $$mathcal{A}_{−}={ f in X : f(t)=−f(−t) space forall t in [−1,1]}. $$



      Is $mathcal{A}_{+} +mathcal{A}_{−} = {f +g : f in mathcal{A}_{+},g in mathcal{A}_{−}}$ meager?




      I know this set is dense by the Stone-Weierstrass Theorem. However, that doesn't really help. I also know that if the set is closed, then it is meager, but I have difficulties deciding whether it is closed or not. I know the exponential function is a limit of a sequence of a sum of even and odd functions, however one could define it to be that, in which case it doesn't help.




      Any hints on how to get going on this problem, and on whether the set $mathcal{A}_{+}+{A}_{-} $ is closed or not? Thank you in advance.






      real-analysis general-topology functional-analysis metric-spaces baire-category




      real-analysis general-topology functional-analysis metric-spaces baire-category






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 2 '18 at 7:14









      Gaby Alfonso

      682315




      682315






















          1 Answer
          1






          active

          oldest

          votes


















          16














          Note that any function can be written as
          $f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
          $mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.



          (It is not meagre because $C[-1,1]$ is a complete metric space.)






          share|cite|improve this answer



















          • 1




            As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
            – R..
            Dec 2 '18 at 14:07










          • @R.. Fourier decomposition is way more complicated than necessary.
            – leftaroundabout
            Dec 2 '18 at 19:05










          • @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
            – R..
            Dec 2 '18 at 21:18










          • @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
            – leftaroundabout
            Dec 2 '18 at 21:50






          • 1




            I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
            – copper.hat
            Dec 2 '18 at 22:36











          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%2f3022345%2fis-the-set-of-continuous-functions-that-are-the-sum-of-even-and-odd-functions-me%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









          16














          Note that any function can be written as
          $f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
          $mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.



          (It is not meagre because $C[-1,1]$ is a complete metric space.)






          share|cite|improve this answer



















          • 1




            As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
            – R..
            Dec 2 '18 at 14:07










          • @R.. Fourier decomposition is way more complicated than necessary.
            – leftaroundabout
            Dec 2 '18 at 19:05










          • @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
            – R..
            Dec 2 '18 at 21:18










          • @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
            – leftaroundabout
            Dec 2 '18 at 21:50






          • 1




            I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
            – copper.hat
            Dec 2 '18 at 22:36
















          16














          Note that any function can be written as
          $f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
          $mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.



          (It is not meagre because $C[-1,1]$ is a complete metric space.)






          share|cite|improve this answer



















          • 1




            As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
            – R..
            Dec 2 '18 at 14:07










          • @R.. Fourier decomposition is way more complicated than necessary.
            – leftaroundabout
            Dec 2 '18 at 19:05










          • @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
            – R..
            Dec 2 '18 at 21:18










          • @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
            – leftaroundabout
            Dec 2 '18 at 21:50






          • 1




            I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
            – copper.hat
            Dec 2 '18 at 22:36














          16












          16








          16






          Note that any function can be written as
          $f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
          $mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.



          (It is not meagre because $C[-1,1]$ is a complete metric space.)






          share|cite|improve this answer














          Note that any function can be written as
          $f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
          $mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.



          (It is not meagre because $C[-1,1]$ is a complete metric space.)







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 2 '18 at 7:23

























          answered Dec 2 '18 at 7:17









          copper.hat

          126k559159




          126k559159








          • 1




            As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
            – R..
            Dec 2 '18 at 14:07










          • @R.. Fourier decomposition is way more complicated than necessary.
            – leftaroundabout
            Dec 2 '18 at 19:05










          • @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
            – R..
            Dec 2 '18 at 21:18










          • @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
            – leftaroundabout
            Dec 2 '18 at 21:50






          • 1




            I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
            – copper.hat
            Dec 2 '18 at 22:36














          • 1




            As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
            – R..
            Dec 2 '18 at 14:07










          • @R.. Fourier decomposition is way more complicated than necessary.
            – leftaroundabout
            Dec 2 '18 at 19:05










          • @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
            – R..
            Dec 2 '18 at 21:18










          • @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
            – leftaroundabout
            Dec 2 '18 at 21:50






          • 1




            I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
            – copper.hat
            Dec 2 '18 at 22:36








          1




          1




          As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
          – R..
          Dec 2 '18 at 14:07




          As an aside, if OP is familiar with Fourier series this decomposition should be part of the associated intuition.
          – R..
          Dec 2 '18 at 14:07












          @R.. Fourier decomposition is way more complicated than necessary.
          – leftaroundabout
          Dec 2 '18 at 19:05




          @R.. Fourier decomposition is way more complicated than necessary.
          – leftaroundabout
          Dec 2 '18 at 19:05












          @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
          – R..
          Dec 2 '18 at 21:18




          @leftaroundabout: Absolutely. copper.hat's answer is simpler and more direct, but having studied Fourier series should leave one with an intuition that there is a natural decomposition into even and odd parts, whereas otherwise this may seem like a surprising and/or obscure result.
          – R..
          Dec 2 '18 at 21:18












          @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
          – leftaroundabout
          Dec 2 '18 at 21:50




          @R.. hm, I rather see it the other way around – this decomposition is one of the foundations that make Fourier analysis possible. But, I suppose this is somewhat a matter of taste.
          – leftaroundabout
          Dec 2 '18 at 21:50




          1




          1




          I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
          – copper.hat
          Dec 2 '18 at 22:36




          I guess I would see the decomposition of square matrices into symmetric and skew symmetric parts a more natural intuition. Fourier series requires a lot more baggage.
          – copper.hat
          Dec 2 '18 at 22:36


















          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%2f3022345%2fis-the-set-of-continuous-functions-that-are-the-sum-of-even-and-odd-functions-me%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

          mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

          Image
          0 I just uploaded a new plugin to my wordpress site and I got this warning: mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924 Usually if I have a problem, I can find answers on google but this one seems pretty particular and I'm not finding any info on it except for this forum. I'm not terribly techie but if someone can tell me how to read this and how to change it I can follow directions pretty well. Thanks All! PS If I don't have the proper tags here could someone add the appropriate tags? php mysql wordpress share | improve this question asked Nov 16 '18 at 20:41 ...
          Read more

          How to change which sound is reproduced for terminal bell?

          Image
          3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
          Read more

          Can I use Tabulator js library in my java Spring + Thymeleaf project?

          Image
          0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
          Read more