Calculating the norm of a specific bounded linear functional












3












$begingroup$


Let $L_1([0,1], m)$ be the Banach space of $mathbb{K}$-valued (i.e. $mathbb{C}$ or $mathbb{R}$-valued) Lebesgue-integrable functions, where $m$ is the Lebesgue measure, be equipped with the norm $‖f‖_1=int_{[0,1]}|f|,dm$ for $f in L_1([0,1], m)$. For $n geq 1$ define $g_n(x) =nsin(n^2x)$, for $x in [0,1]$.
Furthermore, let $phi_n(f)=int_{[0,1]} f g_n , dm$.



I'm then asked to show that $phi_n(f)$ defines a bounded linear map (which I've done) and to show $‖phi_n‖=n$ for $n geq 2$. What I've done is that I have shown $‖phi_n‖leq n$. If I could find a function $f in L_1([0,1], m)$ such that $|phi_n(f)|=n$ and $‖f‖_1=1$, then I would have shown equality. How can I find such a function?



The problem also asks whether I can find a function $f in L_1([0,1], m)$ such that $lim_{n rightarrow infty} |phi_n(f)|=infty$ (perhaps I could use the same function as above?)






functional-analysis lebesgue-measure lp-spaces







share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    Let $L_1([0,1], m)$ be the Banach space of $mathbb{K}$-valued (i.e. $mathbb{C}$ or $mathbb{R}$-valued) Lebesgue-integrable functions, where $m$ is the Lebesgue measure, be equipped with the norm $‖f‖_1=int_{[0,1]}|f|,dm$ for $f in L_1([0,1], m)$. For $n geq 1$ define $g_n(x) =nsin(n^2x)$, for $x in [0,1]$.
    Furthermore, let $phi_n(f)=int_{[0,1]} f g_n , dm$.



    I'm then asked to show that $phi_n(f)$ defines a bounded linear map (which I've done) and to show $‖phi_n‖=n$ for $n geq 2$. What I've done is that I have shown $‖phi_n‖leq n$. If I could find a function $f in L_1([0,1], m)$ such that $|phi_n(f)|=n$ and $‖f‖_1=1$, then I would have shown equality. How can I find such a function?



    The problem also asks whether I can find a function $f in L_1([0,1], m)$ such that $lim_{n rightarrow infty} |phi_n(f)|=infty$ (perhaps I could use the same function as above?)






    functional-analysis lebesgue-measure lp-spaces







    share|cite|improve this question









    $endgroup$















      3












      3








      3


      1



      $begingroup$


      Let $L_1([0,1], m)$ be the Banach space of $mathbb{K}$-valued (i.e. $mathbb{C}$ or $mathbb{R}$-valued) Lebesgue-integrable functions, where $m$ is the Lebesgue measure, be equipped with the norm $‖f‖_1=int_{[0,1]}|f|,dm$ for $f in L_1([0,1], m)$. For $n geq 1$ define $g_n(x) =nsin(n^2x)$, for $x in [0,1]$.
      Furthermore, let $phi_n(f)=int_{[0,1]} f g_n , dm$.



      I'm then asked to show that $phi_n(f)$ defines a bounded linear map (which I've done) and to show $‖phi_n‖=n$ for $n geq 2$. What I've done is that I have shown $‖phi_n‖leq n$. If I could find a function $f in L_1([0,1], m)$ such that $|phi_n(f)|=n$ and $‖f‖_1=1$, then I would have shown equality. How can I find such a function?



      The problem also asks whether I can find a function $f in L_1([0,1], m)$ such that $lim_{n rightarrow infty} |phi_n(f)|=infty$ (perhaps I could use the same function as above?)






      functional-analysis lebesgue-measure lp-spaces







      share|cite|improve this question









      $endgroup$




      Let $L_1([0,1], m)$ be the Banach space of $mathbb{K}$-valued (i.e. $mathbb{C}$ or $mathbb{R}$-valued) Lebesgue-integrable functions, where $m$ is the Lebesgue measure, be equipped with the norm $‖f‖_1=int_{[0,1]}|f|,dm$ for $f in L_1([0,1], m)$. For $n geq 1$ define $g_n(x) =nsin(n^2x)$, for $x in [0,1]$.
      Furthermore, let $phi_n(f)=int_{[0,1]} f g_n , dm$.



      I'm then asked to show that $phi_n(f)$ defines a bounded linear map (which I've done) and to show $‖phi_n‖=n$ for $n geq 2$. What I've done is that I have shown $‖phi_n‖leq n$. If I could find a function $f in L_1([0,1], m)$ such that $|phi_n(f)|=n$ and $‖f‖_1=1$, then I would have shown equality. How can I find such a function?



      The problem also asks whether I can find a function $f in L_1([0,1], m)$ such that $lim_{n rightarrow infty} |phi_n(f)|=infty$ (perhaps I could use the same function as above?)






      functional-analysis lebesgue-measure lp-spaces




      functional-analysis lebesgue-measure lp-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 6 '18 at 22:01









      VercingetorixVercingetorix

      285




      285






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          It is a general fact that if $g$ is a bounded function, then the norm of the linear functional $Lcolon mathbb L^1tomathbb R$ defined by $L(f)=int_{(0,1)}g(x)f(x)mathrm dlambda(x)$ is $leftlVert grightrVert_infty$. Indeed, fix a positive $varepsilon<leftlVert grightrVert_infty$ and define the sets $A^+:=left{xin (0,1), g(x)gt leftlVert grightrVert_infty-varepsilonright}$ and $A^-:=left{xin (0,1), -g(x)gt leftlVert grightrVert_infty-varepsilonright}$. Then
          $$
          maxleft{Lleft(frac 1{lambdaleft(A^+right)}mathbf 1_{A^+}right),Lleft(frac 1{lambdaleft(A^-right)}mathbf 1_{A^-}right) right}geqslant leftlVert grightrVert_infty-varepsilon
          $$
          and (at least one of) the involved function are well-defined and their $L^1$-norm is $1$.



          For the other question, look at one of the functions $xmapsto pm x^{-alpha}$ for $1/2ltalphalt 1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:20












          • $begingroup$
            but I don't know how to do it without explicitly finding a function
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:27










          • $begingroup$
            The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
            $endgroup$
            – Davide Giraudo
            Dec 11 '18 at 19:05










          • $begingroup$
            Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 19:49













          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%2f3029107%2fcalculating-the-norm-of-a-specific-bounded-linear-functional%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









          1












          $begingroup$

          It is a general fact that if $g$ is a bounded function, then the norm of the linear functional $Lcolon mathbb L^1tomathbb R$ defined by $L(f)=int_{(0,1)}g(x)f(x)mathrm dlambda(x)$ is $leftlVert grightrVert_infty$. Indeed, fix a positive $varepsilon<leftlVert grightrVert_infty$ and define the sets $A^+:=left{xin (0,1), g(x)gt leftlVert grightrVert_infty-varepsilonright}$ and $A^-:=left{xin (0,1), -g(x)gt leftlVert grightrVert_infty-varepsilonright}$. Then
          $$
          maxleft{Lleft(frac 1{lambdaleft(A^+right)}mathbf 1_{A^+}right),Lleft(frac 1{lambdaleft(A^-right)}mathbf 1_{A^-}right) right}geqslant leftlVert grightrVert_infty-varepsilon
          $$
          and (at least one of) the involved function are well-defined and their $L^1$-norm is $1$.



          For the other question, look at one of the functions $xmapsto pm x^{-alpha}$ for $1/2ltalphalt 1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:20












          • $begingroup$
            but I don't know how to do it without explicitly finding a function
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:27










          • $begingroup$
            The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
            $endgroup$
            – Davide Giraudo
            Dec 11 '18 at 19:05










          • $begingroup$
            Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 19:49


















          1












          $begingroup$

          It is a general fact that if $g$ is a bounded function, then the norm of the linear functional $Lcolon mathbb L^1tomathbb R$ defined by $L(f)=int_{(0,1)}g(x)f(x)mathrm dlambda(x)$ is $leftlVert grightrVert_infty$. Indeed, fix a positive $varepsilon<leftlVert grightrVert_infty$ and define the sets $A^+:=left{xin (0,1), g(x)gt leftlVert grightrVert_infty-varepsilonright}$ and $A^-:=left{xin (0,1), -g(x)gt leftlVert grightrVert_infty-varepsilonright}$. Then
          $$
          maxleft{Lleft(frac 1{lambdaleft(A^+right)}mathbf 1_{A^+}right),Lleft(frac 1{lambdaleft(A^-right)}mathbf 1_{A^-}right) right}geqslant leftlVert grightrVert_infty-varepsilon
          $$
          and (at least one of) the involved function are well-defined and their $L^1$-norm is $1$.



          For the other question, look at one of the functions $xmapsto pm x^{-alpha}$ for $1/2ltalphalt 1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:20












          • $begingroup$
            but I don't know how to do it without explicitly finding a function
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:27










          • $begingroup$
            The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
            $endgroup$
            – Davide Giraudo
            Dec 11 '18 at 19:05










          • $begingroup$
            Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 19:49
















          1












          1








          1





          $begingroup$

          It is a general fact that if $g$ is a bounded function, then the norm of the linear functional $Lcolon mathbb L^1tomathbb R$ defined by $L(f)=int_{(0,1)}g(x)f(x)mathrm dlambda(x)$ is $leftlVert grightrVert_infty$. Indeed, fix a positive $varepsilon<leftlVert grightrVert_infty$ and define the sets $A^+:=left{xin (0,1), g(x)gt leftlVert grightrVert_infty-varepsilonright}$ and $A^-:=left{xin (0,1), -g(x)gt leftlVert grightrVert_infty-varepsilonright}$. Then
          $$
          maxleft{Lleft(frac 1{lambdaleft(A^+right)}mathbf 1_{A^+}right),Lleft(frac 1{lambdaleft(A^-right)}mathbf 1_{A^-}right) right}geqslant leftlVert grightrVert_infty-varepsilon
          $$
          and (at least one of) the involved function are well-defined and their $L^1$-norm is $1$.



          For the other question, look at one of the functions $xmapsto pm x^{-alpha}$ for $1/2ltalphalt 1$.






          share|cite|improve this answer









          $endgroup$



          It is a general fact that if $g$ is a bounded function, then the norm of the linear functional $Lcolon mathbb L^1tomathbb R$ defined by $L(f)=int_{(0,1)}g(x)f(x)mathrm dlambda(x)$ is $leftlVert grightrVert_infty$. Indeed, fix a positive $varepsilon<leftlVert grightrVert_infty$ and define the sets $A^+:=left{xin (0,1), g(x)gt leftlVert grightrVert_infty-varepsilonright}$ and $A^-:=left{xin (0,1), -g(x)gt leftlVert grightrVert_infty-varepsilonright}$. Then
          $$
          maxleft{Lleft(frac 1{lambdaleft(A^+right)}mathbf 1_{A^+}right),Lleft(frac 1{lambdaleft(A^-right)}mathbf 1_{A^-}right) right}geqslant leftlVert grightrVert_infty-varepsilon
          $$
          and (at least one of) the involved function are well-defined and their $L^1$-norm is $1$.



          For the other question, look at one of the functions $xmapsto pm x^{-alpha}$ for $1/2ltalphalt 1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 10 '18 at 23:04









          Davide GiraudoDavide Giraudo

          127k17154268




          127k17154268












          • $begingroup$
            I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:20












          • $begingroup$
            but I don't know how to do it without explicitly finding a function
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:27










          • $begingroup$
            The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
            $endgroup$
            – Davide Giraudo
            Dec 11 '18 at 19:05










          • $begingroup$
            Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 19:49




















          • $begingroup$
            I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:20












          • $begingroup$
            but I don't know how to do it without explicitly finding a function
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 18:27










          • $begingroup$
            The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
            $endgroup$
            – Davide Giraudo
            Dec 11 '18 at 19:05










          • $begingroup$
            Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
            $endgroup$
            – Vercingetorix
            Dec 11 '18 at 19:49


















          $begingroup$
          I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 18:20






          $begingroup$
          I can see that the first one is correct, but if you calculate $phi_n(x^{-alpha})$ I get $n^{2alpha-1} int_0^{n^2} frac{sin(x)}{x^{alpha}} , dm$ (but it's not that easy to verify that this diverges to infinity}. I also know that we can show the last part without explicitly having to find $f$ (using methods from functional analysis)
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 18:20














          $begingroup$
          but I don't know how to do it without explicitly finding a function
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 18:27




          $begingroup$
          but I don't know how to do it without explicitly finding a function
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 18:27












          $begingroup$
          The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
          $endgroup$
          – Davide Giraudo
          Dec 11 '18 at 19:05




          $begingroup$
          The integral $int_0^{infty}sin x/x^{alpha}$ is convergent. Which methods of functional analysis would you have at your disposal for showing the existence of $f$ without giving a function.
          $endgroup$
          – Davide Giraudo
          Dec 11 '18 at 19:05












          $begingroup$
          Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 19:49






          $begingroup$
          Perhaps the Banach-Steinhaus Theorem? Using the contraposed of Banach-Steinhaus gives that there exist $f in L_1([0,1],m)$ such that $sup_{n in mathbb{N}} |phi_n(f)|=infty$, but I'm not sure where I can go from there to conclude that $|phi_n(f)| rightarrow infty$.
          $endgroup$
          – Vercingetorix
          Dec 11 '18 at 19:49




















          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%2f3029107%2fcalculating-the-norm-of-a-specific-bounded-linear-functional%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

          How to send String Array data to Server using php in android

          Image
          0 I have a String array which contains value string array = {6,37,38,837,7,...} and I have also used some string values. I want to send that all string values and string array to my database which is located at server side. When I am click on submit button the data will stored in table. On the PHP side I want to read this array as it is,other thing i will do at background means PHP side. See my below code then you will get it what I am says. Here is my code: public class Hotels extends Activity { // Declare Variables JSONArray jsonarray = null; private static final String TAG_ID = "id"; public static final String TAG_NAME = "name"; public static final String TAG_LOCATION = "location"; public static final String TAG_DESC = "description"; String f_date, l_date;...
          Read more

          Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

          Image
          up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx...
          Read more

          Is anime1.com a legal site for watching anime?

          Image
          up vote 4 down vote favorite 1 Other anime streaming sites are blocking my IP because I'm Iranian while anime1.com isn't. I wanted to know if it's legal to watch/stream anime from this site. Thanks. anime-production share | improve this question edited Nov 27 at 22:38 W. Are 1,599 2 23 asked Nov 27 at 20:36 mhnajafi 23 3 add a comment ...
          Read more