Mixing
Reference on Lipschitz property of the infimum of a family of Lipschitz functions
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I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.
However, since this is a very basic result, I am interested in a reference where it is proved.
Any suggestions?
functional-analysis reference-request nonlinear-optimization lipschitz-functions non-convex-optimization
asked Jan 31 at 9:49
John DJohn D
1,506624
$endgroup$
add a comment |
$begingroup$
I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.
However, since this is a very basic result, I am interested in a reference where it is proved.
Any suggestions?
functional-analysis reference-request nonlinear-optimization lipschitz-functions non-convex-optimization
asked Jan 31 at 9:49
John DJohn D
1,506624
$endgroup$
$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09
add a comment |
$begingroup$
I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.
However, since this is a very basic result, I am interested in a reference where it is proved.
Any suggestions?
functional-analysis reference-request nonlinear-optimization lipschitz-functions non-convex-optimization
asked Jan 31 at 9:49
John DJohn D
1,506624
$endgroup$
I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.
However, since this is a very basic result, I am interested in a reference where it is proved.
Any suggestions?
functional-analysis reference-request nonlinear-optimization lipschitz-functions non-convex-optimization
functional-analysis reference-request nonlinear-optimization lipschitz-functions non-convex-optimization
asked Jan 31 at 9:49
John DJohn D
1,506624
asked Jan 31 at 9:49
John DJohn D
1,506624
asked Jan 31 at 9:49
John DJohn D
1,506624
asked Jan 31 at 9:49
John DJohn D
1,506624
asked Jan 31 at 9:49
John DJohn D
1,506624
1,506624
$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09
add a comment |
$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09
$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09
$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You may not find this in a text but it is an easy
y consequence of the following: $max{f,g}=frac {f+g+|f-g|} 2$, $min{f,g}=frac {f+g-|f-g|} 2$ and $||a|-|b||leq |a-b|$.
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
$endgroup$
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
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$begingroup$
You may not find this in a text but it is an easy
y consequence of the following: $max{f,g}=frac {f+g+|f-g|} 2$, $min{f,g}=frac {f+g-|f-g|} 2$ and $||a|-|b||leq |a-b|$.
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
$endgroup$
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
add a comment |
$begingroup$
You may not find this in a text but it is an easy
y consequence of the following: $max{f,g}=frac {f+g+|f-g|} 2$, $min{f,g}=frac {f+g-|f-g|} 2$ and $||a|-|b||leq |a-b|$.
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
$endgroup$
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
add a comment |
$begingroup$
You may not find this in a text but it is an easy
y consequence of the following: $max{f,g}=frac {f+g+|f-g|} 2$, $min{f,g}=frac {f+g-|f-g|} 2$ and $||a|-|b||leq |a-b|$.
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
$endgroup$
You may not find this in a text but it is an easy
y consequence of the following: $max{f,g}=frac {f+g+|f-g|} 2$, $min{f,g}=frac {f+g-|f-g|} 2$ and $||a|-|b||leq |a-b|$.
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
answered Jan 31 at 9:54
Kavi Rama MurthyKavi Rama Murthy
72.6k53170
72.6k53170
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
add a comment |
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
$begingroup$
thanks, but I already know how to prove it. What I need is a reference.
$endgroup$
– John D
Jan 31 at 10:01
1
1
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
$begingroup$
@JohnD: If you have a proof, you don't need a reference :)
$endgroup$
– Mundron Schmidt
Jan 31 at 10:02
add a comment |
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$begingroup$
This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple.
$endgroup$
– MaoWao
Jan 31 at 10:09