Isometry between two $L^p$ spaces
I have a measurable space $(X,M,mu)$ and $mu$ is $sigma-$finite, then there exist a finite measure $lambda$ s.t. the space $L^p(mu)$ is isometric to $L^p(lambda)$.
First I proved that there exist a function $h$ in $L(mu)$ which is always positive. Then the problem is how to find the operator, because I can multiply for the inverse of $h(x)$, but how to define this measure? I tried $lambda(A)= int_{A} h(x) dmu$ but then how can I send functions in the first space to the other?
real-analysis functional-analysis measure-theory lp-spaces isometry
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
asked May 10 '17 at 21:13
energy
448
add a comment |
I have a measurable space $(X,M,mu)$ and $mu$ is $sigma-$finite, then there exist a finite measure $lambda$ s.t. the space $L^p(mu)$ is isometric to $L^p(lambda)$.
First I proved that there exist a function $h$ in $L(mu)$ which is always positive. Then the problem is how to find the operator, because I can multiply for the inverse of $h(x)$, but how to define this measure? I tried $lambda(A)= int_{A} h(x) dmu$ but then how can I send functions in the first space to the other?
real-analysis functional-analysis measure-theory lp-spaces isometry
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
asked May 10 '17 at 21:13
energy
448
add a comment |
I have a measurable space $(X,M,mu)$ and $mu$ is $sigma-$finite, then there exist a finite measure $lambda$ s.t. the space $L^p(mu)$ is isometric to $L^p(lambda)$.
First I proved that there exist a function $h$ in $L(mu)$ which is always positive. Then the problem is how to find the operator, because I can multiply for the inverse of $h(x)$, but how to define this measure? I tried $lambda(A)= int_{A} h(x) dmu$ but then how can I send functions in the first space to the other?
real-analysis functional-analysis measure-theory lp-spaces isometry
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
asked May 10 '17 at 21:13
energy
448
I have a measurable space $(X,M,mu)$ and $mu$ is $sigma-$finite, then there exist a finite measure $lambda$ s.t. the space $L^p(mu)$ is isometric to $L^p(lambda)$.
First I proved that there exist a function $h$ in $L(mu)$ which is always positive. Then the problem is how to find the operator, because I can multiply for the inverse of $h(x)$, but how to define this measure? I tried $lambda(A)= int_{A} h(x) dmu$ but then how can I send functions in the first space to the other?
real-analysis functional-analysis measure-theory lp-spaces isometry
real-analysis functional-analysis measure-theory lp-spaces isometry
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
asked May 10 '17 at 21:13
energy
448
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
asked May 10 '17 at 21:13
energy
448
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
edited Dec 11 '18 at 21:04
Davide Giraudo
125k16150260
125k16150260
asked May 10 '17 at 21:13
energy
448
asked May 10 '17 at 21:13
energy
448
asked May 10 '17 at 21:13
energy
448
448
add a comment |
add a comment |
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