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Is there a version of Fisher-Riesz theorem for Banach space?
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$( Omega,F, P )$: a measurable space equiped with a finite measure
$(B , Vert cdot Vert) $ : a Banach space with $mathcalB$ as its borelian $sigma$-algebra
$p$ : a constant bigger than $1$
Define $L^p(Omega, B )$ the vector space that contain all $( F, mathcalB)$-measurable function $f$ such that :
$ vert Vert f Vert vert = sqrt[p] int_Omega Vert f Vert ^p cdot dP < infty$
I'm looking for a version of Riesz-Fischer theorem which affirms that:
Proposition:
$left( L^p(Omega, B ) , vert Vert cdot Vert vert right)$ is a Banach space
With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.
So my question is: " Is the above proposition true? And does anyone have references to this matter?"
banach-spaces integration
asked 6 hours ago
Taro NGUYEN
765
add a comment |Â
up vote
1
down vote
favorite
$( Omega,F, P )$: a measurable space equiped with a finite measure
$(B , Vert cdot Vert) $ : a Banach space with $mathcalB$ as its borelian $sigma$-algebra
$p$ : a constant bigger than $1$
Define $L^p(Omega, B )$ the vector space that contain all $( F, mathcalB)$-measurable function $f$ such that :
$ vert Vert f Vert vert = sqrt[p] int_Omega Vert f Vert ^p cdot dP < infty$
I'm looking for a version of Riesz-Fischer theorem which affirms that:
Proposition:
$left( L^p(Omega, B ) , vert Vert cdot Vert vert right)$ is a Banach space
With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.
So my question is: " Is the above proposition true? And does anyone have references to this matter?"
banach-spaces integration
asked 6 hours ago
Taro NGUYEN
765
1
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
2
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
$( Omega,F, P )$: a measurable space equiped with a finite measure
$(B , Vert cdot Vert) $ : a Banach space with $mathcalB$ as its borelian $sigma$-algebra
$p$ : a constant bigger than $1$
Define $L^p(Omega, B )$ the vector space that contain all $( F, mathcalB)$-measurable function $f$ such that :
$ vert Vert f Vert vert = sqrt[p] int_Omega Vert f Vert ^p cdot dP < infty$
I'm looking for a version of Riesz-Fischer theorem which affirms that:
Proposition:
$left( L^p(Omega, B ) , vert Vert cdot Vert vert right)$ is a Banach space
With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.
So my question is: " Is the above proposition true? And does anyone have references to this matter?"
banach-spaces integration
asked 6 hours ago
Taro NGUYEN
765
$( Omega,F, P )$: a measurable space equiped with a finite measure
$(B , Vert cdot Vert) $ : a Banach space with $mathcalB$ as its borelian $sigma$-algebra
$p$ : a constant bigger than $1$
Define $L^p(Omega, B )$ the vector space that contain all $( F, mathcalB)$-measurable function $f$ such that :
$ vert Vert f Vert vert = sqrt[p] int_Omega Vert f Vert ^p cdot dP < infty$
I'm looking for a version of Riesz-Fischer theorem which affirms that:
Proposition:
$left( L^p(Omega, B ) , vert Vert cdot Vert vert right)$ is a Banach space
With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.
So my question is: " Is the above proposition true? And does anyone have references to this matter?"
banach-spaces integration
banach-spaces integration
asked 6 hours ago
Taro NGUYEN
765
asked 6 hours ago
Taro NGUYEN
765
asked 6 hours ago
Taro NGUYEN
765
asked 6 hours ago
Taro NGUYEN
765
asked 6 hours ago
Taro NGUYEN
765
765
1
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
2
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago
add a comment |Â
1
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
2
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago
1
1
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
2
2
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
These are called Bochner spaces, and yes, they are Banach spaces.
At least one of the standard proofs that $L^p$ is complete goes through basically without change:
Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.
Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.
I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.
answered 6 hours ago
Nate Eldredge
19.2k362108
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
add a comment |Â
up vote
4
down vote
A beautiful treatment of vector valued $L^p$ spaces is in the book:
J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
answered 4 hours ago
Piotr Hajlasz
5,44132053
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
These are called Bochner spaces, and yes, they are Banach spaces.
At least one of the standard proofs that $L^p$ is complete goes through basically without change:
Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.
Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.
I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.
answered 6 hours ago
Nate Eldredge
19.2k362108
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
add a comment |Â
up vote
4
down vote
These are called Bochner spaces, and yes, they are Banach spaces.
At least one of the standard proofs that $L^p$ is complete goes through basically without change:
Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.
Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.
I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.
answered 6 hours ago
Nate Eldredge
19.2k362108
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
add a comment |Â
up vote
4
down vote
up vote
4
down vote
These are called Bochner spaces, and yes, they are Banach spaces.
At least one of the standard proofs that $L^p$ is complete goes through basically without change:
Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.
Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.
I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.
answered 6 hours ago
Nate Eldredge
19.2k362108
These are called Bochner spaces, and yes, they are Banach spaces.
At least one of the standard proofs that $L^p$ is complete goes through basically without change:
Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $||f_n - f_n+1|| le 4^-n$. By Chebyshev's inequality, we then have $P(|f_n - f_n+1| ge 2^-n) le 2^-pn$. Then the Borel-Cantelli lemma implies that for almost every $omega in Omega$, we have $|f_n(omega) - f_n+1(omega)| le 2^-n$ for all but finitely many $n$. In particular, for such $omega$, the sequence $f_n(omega)$ is Cauchy in $B$, so it converges to some $f(omega) in B$.
Now you have that $f$ is the a.e. limit of the $f_n$. Let $epsilon > 0$. Since $f_n$ is Cauchy in $||cdot||$-norm, choose $N$ so large that $||f_n - f_m|| < epsilon$ for all $n,m > N$. Letting $m to infty$ and using Fatou's lemma on the integrals $int_Omega |f_n - f_m|,dP$, conclude that $||f_n - f|| < epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.
I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.
answered 6 hours ago
Nate Eldredge
19.2k362108
edited 5 hours ago
answered 6 hours ago
Nate Eldredge
19.2k362108
answered 6 hours ago
Nate Eldredge
19.2k362108
answered 6 hours ago
Nate Eldredge
19.2k362108
19.2k362108
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
add a comment |Â
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
1
1
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
Just to add a concrete example of quite a recent reference: see for instance "T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach Spaces, Volume I (2016)". Bochner spaces are treated in Chapter I in a rather general setting (for instance, without assuming $sigma$-finiteness in general).
– Jochen Glueck
5 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
I seem to remember some of the basics being sketched in Diestel and Uhl's book Vector Measures
– Yemon Choi
4 hours ago
add a comment |Â
up vote
4
down vote
A beautiful treatment of vector valued $L^p$ spaces is in the book:
J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
answered 4 hours ago
Piotr Hajlasz
5,44132053
add a comment |Â
up vote
4
down vote
A beautiful treatment of vector valued $L^p$ spaces is in the book:
J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
answered 4 hours ago
Piotr Hajlasz
5,44132053
add a comment |Â
up vote
4
down vote
up vote
4
down vote
A beautiful treatment of vector valued $L^p$ spaces is in the book:
J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
answered 4 hours ago
Piotr Hajlasz
5,44132053
A beautiful treatment of vector valued $L^p$ spaces is in the book:
J. Diestel, J. J. Uhl, Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
answered 4 hours ago
Piotr Hajlasz
5,44132053
answered 4 hours ago
Piotr Hajlasz
5,44132053
answered 4 hours ago
Piotr Hajlasz
5,44132053
answered 4 hours ago
Piotr Hajlasz
5,44132053
5,44132053
add a comment |Â
add a comment |Â
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1
You may be interested also in this book springer.com/gp/book/9783540637455
– Tomek Kania
5 hours ago
2
NO! Unless $B$ is separable, or $(Omega, F, P)$ is a special sort of measurable space, this can fail. In general, if you restrict to functions with almost all values in a separable subspace of $B$, then you get the Bochner spaces, which are, indeed, complete.
– Gerald Edgar
4 hours ago
As the issue pointed out by @GeraldEdgar shows, the "right" definition of measurable vector-valued functions is not via the Boral $sigma$-algebra on $B$ but via approximation by simple functions. See for instance Section 1.1 of the book by Hytönen et. al. quoted in my comment below.
– Jochen Glueck
1 hour ago