Extension of a non-negative and symmetric real valued function to a pseudometric












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There exists a result previously stated that shows that a non-negative real valued function $hat{d}:Xtimes Xrightarrow mathbb{R}$ that satisfies symmetry and $d(x,x)=0$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $d(x,y)=0Rightarrow x=y$, then $hat{d}$ can be extended to a metric.






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    There exists a result previously stated that shows that a non-negative real valued function $hat{d}:Xtimes Xrightarrow mathbb{R}$ that satisfies symmetry and $d(x,x)=0$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $d(x,y)=0Rightarrow x=y$, then $hat{d}$ can be extended to a metric.






    metric-spaces







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      There exists a result previously stated that shows that a non-negative real valued function $hat{d}:Xtimes Xrightarrow mathbb{R}$ that satisfies symmetry and $d(x,x)=0$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $d(x,y)=0Rightarrow x=y$, then $hat{d}$ can be extended to a metric.






      metric-spaces







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      There exists a result previously stated that shows that a non-negative real valued function $hat{d}:Xtimes Xrightarrow mathbb{R}$ that satisfies symmetry and $d(x,x)=0$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $d(x,y)=0Rightarrow x=y$, then $hat{d}$ can be extended to a metric.






      metric-spaces




      metric-spaces






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      edited Nov 13 '18 at 5:04









      max_zorn

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      asked Nov 12 '18 at 21:48









      Matheus SilvaMatheus Silva

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          Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $Xtimes X$) understood a function $bar f$ defined on a set $Ysubset X$ (resp. a pseudometric $bar d$ defined on $Ytimes Y$), such that a restriction $bar f|X$ coincides with $f$ (resp. $bar d|Xtimes X=f$).



          I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”



          “The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
          [Bes, Zar]".



          If we have a symmetric non-negative function $d$ on $Xtimes X$ such that $d(x,x)=0$ for each $xin X$, a standard way to modify $d$ to a pseudometric $d’le d$ is to put



          $$d’(x,y)=infleft{sum_{i=1}^{n} d(x_{i-1},x_i):x_1,dots, x_nin X, x_0=x, x_n=yright}.$$



          Remark, that $d’$ may fail to be a metric even when $d(x,y)=0Rightarrow x=y$ for each $x,yin X$.



          References



          [Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.



          [Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.



          [Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.



          [Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.






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            Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $Xtimes X$) understood a function $bar f$ defined on a set $Ysubset X$ (resp. a pseudometric $bar d$ defined on $Ytimes Y$), such that a restriction $bar f|X$ coincides with $f$ (resp. $bar d|Xtimes X=f$).



            I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”



            “The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
            [Bes, Zar]".



            If we have a symmetric non-negative function $d$ on $Xtimes X$ such that $d(x,x)=0$ for each $xin X$, a standard way to modify $d$ to a pseudometric $d’le d$ is to put



            $$d’(x,y)=infleft{sum_{i=1}^{n} d(x_{i-1},x_i):x_1,dots, x_nin X, x_0=x, x_n=yright}.$$



            Remark, that $d’$ may fail to be a metric even when $d(x,y)=0Rightarrow x=y$ for each $x,yin X$.



            References



            [Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.



            [Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.



            [Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.



            [Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.






            share|cite|improve this answer




























              1














              Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $Xtimes X$) understood a function $bar f$ defined on a set $Ysubset X$ (resp. a pseudometric $bar d$ defined on $Ytimes Y$), such that a restriction $bar f|X$ coincides with $f$ (resp. $bar d|Xtimes X=f$).



              I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”



              “The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
              [Bes, Zar]".



              If we have a symmetric non-negative function $d$ on $Xtimes X$ such that $d(x,x)=0$ for each $xin X$, a standard way to modify $d$ to a pseudometric $d’le d$ is to put



              $$d’(x,y)=infleft{sum_{i=1}^{n} d(x_{i-1},x_i):x_1,dots, x_nin X, x_0=x, x_n=yright}.$$



              Remark, that $d’$ may fail to be a metric even when $d(x,y)=0Rightarrow x=y$ for each $x,yin X$.



              References



              [Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.



              [Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.



              [Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.



              [Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.






              share|cite|improve this answer


























                1












                1








                1






                Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $Xtimes X$) understood a function $bar f$ defined on a set $Ysubset X$ (resp. a pseudometric $bar d$ defined on $Ytimes Y$), such that a restriction $bar f|X$ coincides with $f$ (resp. $bar d|Xtimes X=f$).



                I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”



                “The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
                [Bes, Zar]".



                If we have a symmetric non-negative function $d$ on $Xtimes X$ such that $d(x,x)=0$ for each $xin X$, a standard way to modify $d$ to a pseudometric $d’le d$ is to put



                $$d’(x,y)=infleft{sum_{i=1}^{n} d(x_{i-1},x_i):x_1,dots, x_nin X, x_0=x, x_n=yright}.$$



                Remark, that $d’$ may fail to be a metric even when $d(x,y)=0Rightarrow x=y$ for each $x,yin X$.



                References



                [Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.



                [Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.



                [Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.



                [Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.






                share|cite|improve this answer














                Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $Xtimes X$) understood a function $bar f$ defined on a set $Ysubset X$ (resp. a pseudometric $bar d$ defined on $Ytimes Y$), such that a restriction $bar f|X$ coincides with $f$ (resp. $bar d|Xtimes X=f$).



                I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”



                “The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
                [Bes, Zar]".



                If we have a symmetric non-negative function $d$ on $Xtimes X$ such that $d(x,x)=0$ for each $xin X$, a standard way to modify $d$ to a pseudometric $d’le d$ is to put



                $$d’(x,y)=infleft{sum_{i=1}^{n} d(x_{i-1},x_i):x_1,dots, x_nin X, x_0=x, x_n=yright}.$$



                Remark, that $d’$ may fail to be a metric even when $d(x,y)=0Rightarrow x=y$ for each $x,yin X$.



                References



                [Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.



                [Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.



                [Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.



                [Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.







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                edited Jan 4 at 14:10

























                answered Dec 5 '18 at 5:48









                Alex RavskyAlex Ravsky

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