Does this space enjoy Schur property? (proofcheck)











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I need to know if the following reasoning is correct, if there is someone so kind to check it.



Let $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous Sobolev spaces $H^alpha$, $alphainmathbb{R}$, defined in terms of Fourier series by
$$H^alpha = left{f=sum_{kinmathbb{Z}^d} f_k, e_k, f_kinmathbb{C} Big| Vert fVert_{H^alpha}^2:=sum_k vert f_kvert^2(1+vert kvert^2)^alpha<inftyright} $$
This is a family of Hilbert spaces such that $H^alpha$ is compactly embedded in $H^beta$ whenever $alpha>beta$. Now consider the space $H^{-}:=cap_{alpha>0} H^{-alpha}$, endowed with the topology induced by the family of seminorms ${VertcdotVert_{H^{-alpha}}}_{alpha>0}$. Since the $H^{-alpha}$ are contained each one into the other, we can actually only consider a countable amount of them, say $alpha_ndownarrow 0$ and $H^-=cap_n H^{-alpha_n}$, and write an explicit metric which induces the above topology:
$$d(f,g)=sum_n 2^{-n} dfrac{Vert f-gVert_n}{1+Vert f-gVert_n}, quad VertcdotVert_n = VertcdotVert_{H^{-alpha_n}}$$
Now I claim that this space enjoys the Schur property. Let ${f_n}_n$ be a weakly convergent sequence in $H^-$, $f_n rightharpoonup f$, then it must also be weakly convergent in $H^{-alpha_n}$ for each $n$. But since $H^{-alpha_{n+1}} $ is compactly embedded into $H^{-alpha_n}$, weak convergence in the former implies strong convergence in the latter and therefore $f_nto f$ in $H^{-alpha}$ for each $alpha$, which implies strong convergence in $H^-$ as well.



I'd just like to know if this is right since when I first considered this space I was expecting it to be reflexive, but at this point it clearly can't be.










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    I need to know if the following reasoning is correct, if there is someone so kind to check it.



    Let $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous Sobolev spaces $H^alpha$, $alphainmathbb{R}$, defined in terms of Fourier series by
    $$H^alpha = left{f=sum_{kinmathbb{Z}^d} f_k, e_k, f_kinmathbb{C} Big| Vert fVert_{H^alpha}^2:=sum_k vert f_kvert^2(1+vert kvert^2)^alpha<inftyright} $$
    This is a family of Hilbert spaces such that $H^alpha$ is compactly embedded in $H^beta$ whenever $alpha>beta$. Now consider the space $H^{-}:=cap_{alpha>0} H^{-alpha}$, endowed with the topology induced by the family of seminorms ${VertcdotVert_{H^{-alpha}}}_{alpha>0}$. Since the $H^{-alpha}$ are contained each one into the other, we can actually only consider a countable amount of them, say $alpha_ndownarrow 0$ and $H^-=cap_n H^{-alpha_n}$, and write an explicit metric which induces the above topology:
    $$d(f,g)=sum_n 2^{-n} dfrac{Vert f-gVert_n}{1+Vert f-gVert_n}, quad VertcdotVert_n = VertcdotVert_{H^{-alpha_n}}$$
    Now I claim that this space enjoys the Schur property. Let ${f_n}_n$ be a weakly convergent sequence in $H^-$, $f_n rightharpoonup f$, then it must also be weakly convergent in $H^{-alpha_n}$ for each $n$. But since $H^{-alpha_{n+1}} $ is compactly embedded into $H^{-alpha_n}$, weak convergence in the former implies strong convergence in the latter and therefore $f_nto f$ in $H^{-alpha}$ for each $alpha$, which implies strong convergence in $H^-$ as well.



    I'd just like to know if this is right since when I first considered this space I was expecting it to be reflexive, but at this point it clearly can't be.










    share|cite|improve this question
























      up vote
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      favorite









      up vote
      0
      down vote

      favorite











      I need to know if the following reasoning is correct, if there is someone so kind to check it.



      Let $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous Sobolev spaces $H^alpha$, $alphainmathbb{R}$, defined in terms of Fourier series by
      $$H^alpha = left{f=sum_{kinmathbb{Z}^d} f_k, e_k, f_kinmathbb{C} Big| Vert fVert_{H^alpha}^2:=sum_k vert f_kvert^2(1+vert kvert^2)^alpha<inftyright} $$
      This is a family of Hilbert spaces such that $H^alpha$ is compactly embedded in $H^beta$ whenever $alpha>beta$. Now consider the space $H^{-}:=cap_{alpha>0} H^{-alpha}$, endowed with the topology induced by the family of seminorms ${VertcdotVert_{H^{-alpha}}}_{alpha>0}$. Since the $H^{-alpha}$ are contained each one into the other, we can actually only consider a countable amount of them, say $alpha_ndownarrow 0$ and $H^-=cap_n H^{-alpha_n}$, and write an explicit metric which induces the above topology:
      $$d(f,g)=sum_n 2^{-n} dfrac{Vert f-gVert_n}{1+Vert f-gVert_n}, quad VertcdotVert_n = VertcdotVert_{H^{-alpha_n}}$$
      Now I claim that this space enjoys the Schur property. Let ${f_n}_n$ be a weakly convergent sequence in $H^-$, $f_n rightharpoonup f$, then it must also be weakly convergent in $H^{-alpha_n}$ for each $n$. But since $H^{-alpha_{n+1}} $ is compactly embedded into $H^{-alpha_n}$, weak convergence in the former implies strong convergence in the latter and therefore $f_nto f$ in $H^{-alpha}$ for each $alpha$, which implies strong convergence in $H^-$ as well.



      I'd just like to know if this is right since when I first considered this space I was expecting it to be reflexive, but at this point it clearly can't be.










      share|cite|improve this question













      I need to know if the following reasoning is correct, if there is someone so kind to check it.



      Let $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous Sobolev spaces $H^alpha$, $alphainmathbb{R}$, defined in terms of Fourier series by
      $$H^alpha = left{f=sum_{kinmathbb{Z}^d} f_k, e_k, f_kinmathbb{C} Big| Vert fVert_{H^alpha}^2:=sum_k vert f_kvert^2(1+vert kvert^2)^alpha<inftyright} $$
      This is a family of Hilbert spaces such that $H^alpha$ is compactly embedded in $H^beta$ whenever $alpha>beta$. Now consider the space $H^{-}:=cap_{alpha>0} H^{-alpha}$, endowed with the topology induced by the family of seminorms ${VertcdotVert_{H^{-alpha}}}_{alpha>0}$. Since the $H^{-alpha}$ are contained each one into the other, we can actually only consider a countable amount of them, say $alpha_ndownarrow 0$ and $H^-=cap_n H^{-alpha_n}$, and write an explicit metric which induces the above topology:
      $$d(f,g)=sum_n 2^{-n} dfrac{Vert f-gVert_n}{1+Vert f-gVert_n}, quad VertcdotVert_n = VertcdotVert_{H^{-alpha_n}}$$
      Now I claim that this space enjoys the Schur property. Let ${f_n}_n$ be a weakly convergent sequence in $H^-$, $f_n rightharpoonup f$, then it must also be weakly convergent in $H^{-alpha_n}$ for each $n$. But since $H^{-alpha_{n+1}} $ is compactly embedded into $H^{-alpha_n}$, weak convergence in the former implies strong convergence in the latter and therefore $f_nto f$ in $H^{-alpha}$ for each $alpha$, which implies strong convergence in $H^-$ as well.



      I'd just like to know if this is right since when I first considered this space I was expecting it to be reflexive, but at this point it clearly can't be.







      general-topology functional-analysis sobolev-spaces






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      asked Nov 14 at 21:10









      Lucio

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