Extension of a non-negative and symmetric real valued function to a pseudometric
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
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
add a comment |
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
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
add a comment |
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
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
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
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
edited Nov 13 '18 at 5:04
max_zorn
3,29361328
3,29361328
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
asked Nov 12 '18 at 21:48
Matheus SilvaMatheus Silva
306
306
add a comment |
add a comment |
1 Answer
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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.
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
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1 Answer
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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.
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
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.
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
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.
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
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.
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
edited Jan 4 at 14:10
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
answered Dec 5 '18 at 5:48
Alex RavskyAlex Ravsky
39.4k32181
39.4k32181
add a comment |
add a comment |
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