Does the norm $mathcalH := | w(x,y)log^+ (w(x,y)) |_L^1$ define a reflexive Banach space?

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Suppose $Omega subset mathbbR^2$ is an open bounded set. Given the norm
beginequation*
| w(x,y) |_mathcalH(Omega) := | w(x,y) log^+(w(x,y)) |_L^1(Omega times Omega)
endequation*
where
beginequation*
log^+(x) = left{ beginarrayll log(x) & log(x) geq 0 \
0 & textelse endarray right.
endequation*
is the space
beginequation*
mathcalW = left w
endequation*
a reflexive Banach space? I saw this result utilized in a paper without any justification so I just wanted to verify that it is indeed true. Thanks for the help.




functional-analysis analysis banach-spaces




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  • In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
    – Martin Argerami
    Jul 30 at 17:24










  • The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
    – user357151
    Jul 30 at 19:39














up vote
0
down vote

favorite












Suppose $Omega subset mathbbR^2$ is an open bounded set. Given the norm
beginequation*
| w(x,y) |_mathcalH(Omega) := | w(x,y) log^+(w(x,y)) |_L^1(Omega times Omega)
endequation*
where
beginequation*
log^+(x) = left{ beginarrayll log(x) & log(x) geq 0 \
0 & textelse endarray right.
endequation*
is the space
beginequation*
mathcalW = left w
endequation*
a reflexive Banach space? I saw this result utilized in a paper without any justification so I just wanted to verify that it is indeed true. Thanks for the help.




functional-analysis analysis banach-spaces




share|cite|improve this question





















  • In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
    – Martin Argerami
    Jul 30 at 17:24










  • The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
    – user357151
    Jul 30 at 19:39












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose $Omega subset mathbbR^2$ is an open bounded set. Given the norm
beginequation*
| w(x,y) |_mathcalH(Omega) := | w(x,y) log^+(w(x,y)) |_L^1(Omega times Omega)
endequation*
where
beginequation*
log^+(x) = left{ beginarrayll log(x) & log(x) geq 0 \
0 & textelse endarray right.
endequation*
is the space
beginequation*
mathcalW = left w
endequation*
a reflexive Banach space? I saw this result utilized in a paper without any justification so I just wanted to verify that it is indeed true. Thanks for the help.




functional-analysis analysis banach-spaces




share|cite|improve this question













Suppose $Omega subset mathbbR^2$ is an open bounded set. Given the norm
beginequation*
| w(x,y) |_mathcalH(Omega) := | w(x,y) log^+(w(x,y)) |_L^1(Omega times Omega)
endequation*
where
beginequation*
log^+(x) = left{ beginarrayll log(x) & log(x) geq 0 \
0 & textelse endarray right.
endequation*
is the space
beginequation*
mathcalW = left w
endequation*
a reflexive Banach space? I saw this result utilized in a paper without any justification so I just wanted to verify that it is indeed true. Thanks for the help.





functional-analysis analysis banach-spaces





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 30 at 19:18









Martin Argerami

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115k1071164









asked Jul 30 at 16:47









John

1588




1588











  • In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
    – Martin Argerami
    Jul 30 at 17:24










  • The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
    – user357151
    Jul 30 at 19:39
















  • In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
    – Martin Argerami
    Jul 30 at 17:24










  • The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
    – user357151
    Jul 30 at 19:39















In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
– Martin Argerami
Jul 30 at 17:24




In what sense is $mathcal W$ a "space"? It is not a vector space over $mathbb R$ nor $mathbb C$.
– Martin Argerami
Jul 30 at 17:24












The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
– user357151
Jul 30 at 19:39




The quantity you stated isn't a norm. Did you mean to define an Orlicz class? A reference to the paper where you saw this claim is an essential missing part of this question.
– user357151
Jul 30 at 19:39















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