Any example of equivalent norms in infinite-dimensional vector space?












-1












$begingroup$


I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?






vector-spaces normed-spaces







share|cite|improve this question











$endgroup$












  • $begingroup$
    Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
    $endgroup$
    – Zachary Selk
    Dec 25 '18 at 16:19


















-1












$begingroup$


I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?






vector-spaces normed-spaces







share|cite|improve this question











$endgroup$












  • $begingroup$
    Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
    $endgroup$
    – Zachary Selk
    Dec 25 '18 at 16:19
















-1












-1








-1





$begingroup$


I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?






vector-spaces normed-spaces







share|cite|improve this question











$endgroup$




I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?






vector-spaces normed-spaces




vector-spaces normed-spaces






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share|cite|improve this question













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edited Dec 25 '18 at 16:26









Bernard

121k740116




121k740116










asked Dec 25 '18 at 16:14









zeraoulia rafikzeraoulia rafik

2,40811030




2,40811030












  • $begingroup$
    Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
    $endgroup$
    – Zachary Selk
    Dec 25 '18 at 16:19




















  • $begingroup$
    Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
    $endgroup$
    – Zachary Selk
    Dec 25 '18 at 16:19


















$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– Zachary Selk
Dec 25 '18 at 16:19






$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– Zachary Selk
Dec 25 '18 at 16:19












2 Answers
2






active

oldest

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0












$begingroup$

If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
    $$ clVert frVert'le lVert frVertle ClVert frVert'.$$
    For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

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      active

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      active

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      0












      $begingroup$

      If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.






          share|cite|improve this answer









          $endgroup$



          If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 25 '18 at 16:17









          José Carlos SantosJosé Carlos Santos

          163k22131234




          163k22131234























              0












              $begingroup$

              Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
              $$ clVert frVert'le lVert frVertle ClVert frVert'.$$
              For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
                $$ clVert frVert'le lVert frVertle ClVert frVert'.$$
                For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
                  $$ clVert frVert'le lVert frVertle ClVert frVert'.$$
                  For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.






                  share|cite|improve this answer









                  $endgroup$



                  Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
                  $$ clVert frVert'le lVert frVertle ClVert frVert'.$$
                  For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 25 '18 at 16:19









                  Antonios-Alexandros RobotisAntonios-Alexandros Robotis

                  10.4k41641




                  10.4k41641






























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