Find a sequence Cauchy in $L^p(mathbb{R})$, but not Cauchy in $L^q(mathbb{R})$ for $pnot = q$
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For all $p,,qin[1,infty)$, $p not = q$, find a sequence of Lebesgue measurable function $f_n:mathbb{R}rightarrowmathbb{R}$ so that $f_nin cap_{rin[1,infty)}L^r(mathbb{R})$, and $(f_n)_{ninmathbb{N}}$ is a Cauchy sequence in $L^p(mathbb{R})$, but not a Cauchy sequence in $L^q(mathbb{R})$.
I was able to this problem given $p < q$, using result in here, but I cannot find one that true on $p not = q$.
real-analysis
asked Nov 15 at 1:49
Awoo
448
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up vote
0
down vote
favorite
For all $p,,qin[1,infty)$, $p not = q$, find a sequence of Lebesgue measurable function $f_n:mathbb{R}rightarrowmathbb{R}$ so that $f_nin cap_{rin[1,infty)}L^r(mathbb{R})$, and $(f_n)_{ninmathbb{N}}$ is a Cauchy sequence in $L^p(mathbb{R})$, but not a Cauchy sequence in $L^q(mathbb{R})$.
I was able to this problem given $p < q$, using result in here, but I cannot find one that true on $p not = q$.
real-analysis
asked Nov 15 at 1:49
Awoo
448
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For all $p,,qin[1,infty)$, $p not = q$, find a sequence of Lebesgue measurable function $f_n:mathbb{R}rightarrowmathbb{R}$ so that $f_nin cap_{rin[1,infty)}L^r(mathbb{R})$, and $(f_n)_{ninmathbb{N}}$ is a Cauchy sequence in $L^p(mathbb{R})$, but not a Cauchy sequence in $L^q(mathbb{R})$.
I was able to this problem given $p < q$, using result in here, but I cannot find one that true on $p not = q$.
real-analysis
asked Nov 15 at 1:49
Awoo
448
For all $p,,qin[1,infty)$, $p not = q$, find a sequence of Lebesgue measurable function $f_n:mathbb{R}rightarrowmathbb{R}$ so that $f_nin cap_{rin[1,infty)}L^r(mathbb{R})$, and $(f_n)_{ninmathbb{N}}$ is a Cauchy sequence in $L^p(mathbb{R})$, but not a Cauchy sequence in $L^q(mathbb{R})$.
I was able to this problem given $p < q$, using result in here, but I cannot find one that true on $p not = q$.
real-analysis
real-analysis
asked Nov 15 at 1:49
Awoo
448
asked Nov 15 at 1:49
Awoo
448
asked Nov 15 at 1:49
Awoo
448
asked Nov 15 at 1:49
Awoo
448
asked Nov 15 at 1:49
Awoo
448
448
add a comment |
add a comment |
1 Answer
1
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1
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Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.
answered Nov 15 at 1:58
Robert Israel
313k23206452
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.
answered Nov 15 at 1:58
Robert Israel
313k23206452
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
add a comment |
up vote
1
down vote
Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.
answered Nov 15 at 1:58
Robert Israel
313k23206452
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
add a comment |
up vote
1
down vote
up vote
1
down vote
Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.
answered Nov 15 at 1:58
Robert Israel
313k23206452
Hint: try some multiples of indicator functions of intervals. Depending on whether $p < q$ or the reverse, make the graphs tall and skinny or short and fat.
answered Nov 15 at 1:58
Robert Israel
313k23206452
answered Nov 15 at 1:58
Robert Israel
313k23206452
answered Nov 15 at 1:58
Robert Israel
313k23206452
answered Nov 15 at 1:58
Robert Israel
313k23206452
313k23206452
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
add a comment |
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
I can find one by using multiples of indicator functions, but that only works for $p<q$ or reverse. But I want to find one that is true whenever $pnot = q$.
– Awoo
Nov 15 at 5:13
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
You can combine one for $q < p$ and one for $q > p$ to get a sequence that is Cauchy in $L^p$ but not in $L^q$ for any $q ne p$.
– Robert Israel
Nov 15 at 13:50
add a comment |
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