Showing $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is not...












1












$begingroup$


Let $f_t(x):=f(x+t)$



I want to show that $f mapsto f_t$ is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$



$f mapsto f_t$ is obviously linear and bijective, the inverse is $f mapsto f_{-t}$. Because of the translation-invariance of $lambda_p$ we have $|f_t|_p=|f|_p$ and therefore the mapping is a linear, bijective isomorphism.




How can I show that for arbitrary $f in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is not continuous?







real-analysis complex-analysis functional-analysis lp-spaces







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    1












    $begingroup$


    Let $f_t(x):=f(x+t)$



    I want to show that $f mapsto f_t$ is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$



    $f mapsto f_t$ is obviously linear and bijective, the inverse is $f mapsto f_{-t}$. Because of the translation-invariance of $lambda_p$ we have $|f_t|_p=|f|_p$ and therefore the mapping is a linear, bijective isomorphism.




    How can I show that for arbitrary $f in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is not continuous?







    real-analysis complex-analysis functional-analysis lp-spaces







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $f_t(x):=f(x+t)$



      I want to show that $f mapsto f_t$ is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$



      $f mapsto f_t$ is obviously linear and bijective, the inverse is $f mapsto f_{-t}$. Because of the translation-invariance of $lambda_p$ we have $|f_t|_p=|f|_p$ and therefore the mapping is a linear, bijective isomorphism.




      How can I show that for arbitrary $f in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is not continuous?







      real-analysis complex-analysis functional-analysis lp-spaces







      share|cite|improve this question









      $endgroup$




      Let $f_t(x):=f(x+t)$



      I want to show that $f mapsto f_t$ is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$



      $f mapsto f_t$ is obviously linear and bijective, the inverse is $f mapsto f_{-t}$. Because of the translation-invariance of $lambda_p$ we have $|f_t|_p=|f|_p$ and therefore the mapping is a linear, bijective isomorphism.




      How can I show that for arbitrary $f in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is not continuous?







      real-analysis complex-analysis functional-analysis lp-spaces




      real-analysis complex-analysis functional-analysis lp-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 31 '18 at 14:06









      user626880user626880

      204




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          1 Answer
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          $begingroup$

          Consider $p=1$ and $f$ the indicator of the unit interval. Then $leftlVert f_t-f_srightrVert_infty=1$ for all distinct real numbers $s$ and $t$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
            $endgroup$
            – user626880
            Jan 5 at 10:08










          • $begingroup$
            Yes, I have. Maybe you could ask it as a separated question.
            $endgroup$
            – Davide Giraudo
            Jan 5 at 12:47










          • $begingroup$
            link
            $endgroup$
            – user626880
            Jan 5 at 13:57











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          1 Answer
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          1 Answer
          1






          active

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          active

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          active

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          2












          $begingroup$

          Consider $p=1$ and $f$ the indicator of the unit interval. Then $leftlVert f_t-f_srightrVert_infty=1$ for all distinct real numbers $s$ and $t$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
            $endgroup$
            – user626880
            Jan 5 at 10:08










          • $begingroup$
            Yes, I have. Maybe you could ask it as a separated question.
            $endgroup$
            – Davide Giraudo
            Jan 5 at 12:47










          • $begingroup$
            link
            $endgroup$
            – user626880
            Jan 5 at 13:57
















          2












          $begingroup$

          Consider $p=1$ and $f$ the indicator of the unit interval. Then $leftlVert f_t-f_srightrVert_infty=1$ for all distinct real numbers $s$ and $t$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
            $endgroup$
            – user626880
            Jan 5 at 10:08










          • $begingroup$
            Yes, I have. Maybe you could ask it as a separated question.
            $endgroup$
            – Davide Giraudo
            Jan 5 at 12:47










          • $begingroup$
            link
            $endgroup$
            – user626880
            Jan 5 at 13:57














          2












          2








          2





          $begingroup$

          Consider $p=1$ and $f$ the indicator of the unit interval. Then $leftlVert f_t-f_srightrVert_infty=1$ for all distinct real numbers $s$ and $t$.






          share|cite|improve this answer









          $endgroup$



          Consider $p=1$ and $f$ the indicator of the unit interval. Then $leftlVert f_t-f_srightrVert_infty=1$ for all distinct real numbers $s$ and $t$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 31 '18 at 14:24









          Davide GiraudoDavide Giraudo

          128k17154268




          128k17154268












          • $begingroup$
            Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
            $endgroup$
            – user626880
            Jan 5 at 10:08










          • $begingroup$
            Yes, I have. Maybe you could ask it as a separated question.
            $endgroup$
            – Davide Giraudo
            Jan 5 at 12:47










          • $begingroup$
            link
            $endgroup$
            – user626880
            Jan 5 at 13:57


















          • $begingroup$
            Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
            $endgroup$
            – user626880
            Jan 5 at 10:08










          • $begingroup$
            Yes, I have. Maybe you could ask it as a separated question.
            $endgroup$
            – Davide Giraudo
            Jan 5 at 12:47










          • $begingroup$
            link
            $endgroup$
            – user626880
            Jan 5 at 13:57
















          $begingroup$
          Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
          $endgroup$
          – user626880
          Jan 5 at 10:08




          $begingroup$
          Thanks a lot, that works. Do you also have an idea on how to show for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous
          $endgroup$
          – user626880
          Jan 5 at 10:08












          $begingroup$
          Yes, I have. Maybe you could ask it as a separated question.
          $endgroup$
          – Davide Giraudo
          Jan 5 at 12:47




          $begingroup$
          Yes, I have. Maybe you could ask it as a separated question.
          $endgroup$
          – Davide Giraudo
          Jan 5 at 12:47












          $begingroup$
          link
          $endgroup$
          – user626880
          Jan 5 at 13:57




          $begingroup$
          link
          $endgroup$
          – user626880
          Jan 5 at 13:57


















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