Metrics on families of functions












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Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.




I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.




*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.






general-topology functional-analysis metric-spaces big-list







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    $begingroup$
    The Sobolev metrics certainly form an important class of examples.
    $endgroup$
    – pseudocydonia
    Oct 29 '18 at 3:17
















3












$begingroup$


Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.




I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.




*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.






general-topology functional-analysis metric-spaces big-list







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The Sobolev metrics certainly form an important class of examples.
    $endgroup$
    – pseudocydonia
    Oct 29 '18 at 3:17














3












3








3





$begingroup$


Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.




I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.




*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.






general-topology functional-analysis metric-spaces big-list







share|cite|improve this question











$endgroup$




Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.




I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.




*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.






general-topology functional-analysis metric-spaces big-list




general-topology functional-analysis metric-spaces big-list






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edited Dec 6 '18 at 15:18







Anguepa

















asked Oct 16 '18 at 15:16









AnguepaAnguepa

1,406819




1,406819








  • 1




    $begingroup$
    The Sobolev metrics certainly form an important class of examples.
    $endgroup$
    – pseudocydonia
    Oct 29 '18 at 3:17














  • 1




    $begingroup$
    The Sobolev metrics certainly form an important class of examples.
    $endgroup$
    – pseudocydonia
    Oct 29 '18 at 3:17








1




1




$begingroup$
The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17




$begingroup$
The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17










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$begingroup$

There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as



$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$



Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.






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    $begingroup$

    There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as



    $$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$



    Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as



      $$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$



      Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as



        $$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$



        Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.






        share|cite|improve this answer









        $endgroup$



        There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as



        $$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$



        Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 6 '18 at 20:48









        postmortespostmortes

        1,89321117




        1,89321117






























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