Inclusion relationship between $mathscr l^p$ spaces
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In Folland’s Real Analysis book, we have the following proposition:
“ If $mu(X)< infty$ and $0<p<qle infty$, then $L^p(mu)supset L^q(mu)$.”
So suppose that $A$ is a finite set and $mu$ is the counting measure on $A$. If $0<p<qleinfty$, does it follow that $mathscr l^p(A)supset mathscr l^q(A)$? I am asking this question because usually I see the reverse inclusion for $mathscr l^p$ spaces.
real-analysis functional-analysis measure-theory lp-spaces
asked Jan 16 at 16:46
User12239User12239
367216
$endgroup$
add a comment |
$begingroup$
In Folland’s Real Analysis book, we have the following proposition:
“ If $mu(X)< infty$ and $0<p<qle infty$, then $L^p(mu)supset L^q(mu)$.”
So suppose that $A$ is a finite set and $mu$ is the counting measure on $A$. If $0<p<qleinfty$, does it follow that $mathscr l^p(A)supset mathscr l^q(A)$? I am asking this question because usually I see the reverse inclusion for $mathscr l^p$ spaces.
real-analysis functional-analysis measure-theory lp-spaces
asked Jan 16 at 16:46
User12239User12239
367216
$endgroup$
4
$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
1
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03
add a comment |
$begingroup$
In Folland’s Real Analysis book, we have the following proposition:
“ If $mu(X)< infty$ and $0<p<qle infty$, then $L^p(mu)supset L^q(mu)$.”
So suppose that $A$ is a finite set and $mu$ is the counting measure on $A$. If $0<p<qleinfty$, does it follow that $mathscr l^p(A)supset mathscr l^q(A)$? I am asking this question because usually I see the reverse inclusion for $mathscr l^p$ spaces.
real-analysis functional-analysis measure-theory lp-spaces
asked Jan 16 at 16:46
User12239User12239
367216
$endgroup$
In Folland’s Real Analysis book, we have the following proposition:
“ If $mu(X)< infty$ and $0<p<qle infty$, then $L^p(mu)supset L^q(mu)$.”
So suppose that $A$ is a finite set and $mu$ is the counting measure on $A$. If $0<p<qleinfty$, does it follow that $mathscr l^p(A)supset mathscr l^q(A)$? I am asking this question because usually I see the reverse inclusion for $mathscr l^p$ spaces.
real-analysis functional-analysis measure-theory lp-spaces
real-analysis functional-analysis measure-theory lp-spaces
asked Jan 16 at 16:46
User12239User12239
367216
asked Jan 16 at 16:46
User12239User12239
367216
asked Jan 16 at 16:46
User12239User12239
367216
asked Jan 16 at 16:46
User12239User12239
367216
asked Jan 16 at 16:46
User12239User12239
367216
367216
4
$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
1
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03
add a comment |
4
$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
1
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03
4
4
$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
1
1
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03
add a comment |
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$begingroup$
If $A=Bbb N$, then the reverse inclusion is right. But if $A$ is finite, all $l^p$ spaces are the same since finite sum always converges.
$endgroup$
– Song
Jan 16 at 16:48
$begingroup$
thanks I got it @Song
$endgroup$
– User12239
Jan 16 at 16:51
1
$begingroup$
Folland Problem 6.5 I believe characterizes the inclusions of $L^p$ spaces when only one measure is involved, and for the inclusion $L^p(mu) subset L^q(nu)$, Folland references a paper in the Notes and Remarks section of chapter 6.
$endgroup$
– LinearOperator32
Jan 17 at 6:03