Dual spaces of uniformly convex Banach spaces












2












$begingroup$


I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a uniformly convex dual space $X^*$, can anything be said about the uniform convexity of $X$?



I would like links to any papers discussing these points, or any counterexamples. Thank you.






functional-analysis reference-request banach-spaces







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a uniformly convex dual space $X^*$, can anything be said about the uniform convexity of $X$?



    I would like links to any papers discussing these points, or any counterexamples. Thank you.






    functional-analysis reference-request banach-spaces







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a uniformly convex dual space $X^*$, can anything be said about the uniform convexity of $X$?



      I would like links to any papers discussing these points, or any counterexamples. Thank you.






      functional-analysis reference-request banach-spaces







      share|cite|improve this question









      $endgroup$




      I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a uniformly convex dual space $X^*$, can anything be said about the uniform convexity of $X$?



      I would like links to any papers discussing these points, or any counterexamples. Thank you.






      functional-analysis reference-request banach-spaces




      functional-analysis reference-request banach-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 16 '18 at 12:54









      talfredtalfred

      756




      756






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.



          EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.



          Example: The norm $|cdot|_1+|cdot|_2$ is uniformly convex, but not uniformly smooth if $dim Xgeq2$. Hence its dual is uniformly smooth, but not uniformly convex.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:09










          • $begingroup$
            Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:10






          • 1




            $begingroup$
            I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:12








          • 2




            $begingroup$
            Can't argue with that. I edited the post.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:13











          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%2f3042581%2fdual-spaces-of-uniformly-convex-banach-spaces%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









          3












          $begingroup$

          A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.



          EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.



          Example: The norm $|cdot|_1+|cdot|_2$ is uniformly convex, but not uniformly smooth if $dim Xgeq2$. Hence its dual is uniformly smooth, but not uniformly convex.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:09










          • $begingroup$
            Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:10






          • 1




            $begingroup$
            I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:12








          • 2




            $begingroup$
            Can't argue with that. I edited the post.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:13
















          3












          $begingroup$

          A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.



          EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.



          Example: The norm $|cdot|_1+|cdot|_2$ is uniformly convex, but not uniformly smooth if $dim Xgeq2$. Hence its dual is uniformly smooth, but not uniformly convex.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:09










          • $begingroup$
            Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:10






          • 1




            $begingroup$
            I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:12








          • 2




            $begingroup$
            Can't argue with that. I edited the post.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:13














          3












          3








          3





          $begingroup$

          A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.



          EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.



          Example: The norm $|cdot|_1+|cdot|_2$ is uniformly convex, but not uniformly smooth if $dim Xgeq2$. Hence its dual is uniformly smooth, but not uniformly convex.






          share|cite|improve this answer











          $endgroup$



          A Banach space is uniformly convex if and only if its dual is uniformly smooth. Also, since any uniformly convex Banach space is reflexive, this goes both ways, and any uniformly smooth Banach space is also reflexive. This can all be found on the Wikipedia articles for these properties, and they contain references at the bottom.



          EDIT: Because uniformly convex and uniformly smooth are independent, we can find Banach spaces that are uniformly convex but which have a dual space that is not, and vise versa. Same for uniformly smooth.



          Example: The norm $|cdot|_1+|cdot|_2$ is uniformly convex, but not uniformly smooth if $dim Xgeq2$. Hence its dual is uniformly smooth, but not uniformly convex.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 16 '18 at 13:12

























          answered Dec 16 '18 at 13:02









          SmileyCraftSmileyCraft

          3,401516




          3,401516








          • 1




            $begingroup$
            "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:09










          • $begingroup$
            Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:10






          • 1




            $begingroup$
            I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:12








          • 2




            $begingroup$
            Can't argue with that. I edited the post.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:13














          • 1




            $begingroup$
            "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:09










          • $begingroup$
            Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:10






          • 1




            $begingroup$
            I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
            $endgroup$
            – Theo Bendit
            Dec 16 '18 at 13:12








          • 2




            $begingroup$
            Can't argue with that. I edited the post.
            $endgroup$
            – SmileyCraft
            Dec 16 '18 at 13:13








          1




          1




          $begingroup$
          "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
          $endgroup$
          – Theo Bendit
          Dec 16 '18 at 13:09




          $begingroup$
          "Because uniformly convex and uniformly smooth are mutually exclusive..." What about Hilbert Spaces?
          $endgroup$
          – Theo Bendit
          Dec 16 '18 at 13:09












          $begingroup$
          Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
          $endgroup$
          – SmileyCraft
          Dec 16 '18 at 13:10




          $begingroup$
          Mutually exclusive means any combination of the two can appear. Hilbert spaces are just an example where both properties are held. The norm $|cdot|_p$ for $pin{1,infty}$ is neither, and I already gave examples where either one is held, while the other is not.
          $endgroup$
          – SmileyCraft
          Dec 16 '18 at 13:10




          1




          1




          $begingroup$
          I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
          $endgroup$
          – Theo Bendit
          Dec 16 '18 at 13:12






          $begingroup$
          I think you mean "independent" in that case. Mutually exclusive means that both cannot simultaneously occur.
          $endgroup$
          – Theo Bendit
          Dec 16 '18 at 13:12






          2




          2




          $begingroup$
          Can't argue with that. I edited the post.
          $endgroup$
          – SmileyCraft
          Dec 16 '18 at 13:13




          $begingroup$
          Can't argue with that. I edited the post.
          $endgroup$
          – SmileyCraft
          Dec 16 '18 at 13:13


















          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%2f3042581%2fdual-spaces-of-uniformly-convex-banach-spaces%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more