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Is every Banach space embedded in a reflexive space?

Is every Banach space densely embedded in a Hilbert space?reflexive Banach spaceIs there a non-reflexive Banach space with every proper closed subspace reflexive?A Banach space is reflexive if a closed subspace and its quotient space are both reflexiveClosed subspace of a reflexive Banach space is reflexiveEquivalent characterizations for Reflexive spacesIs any closed and bounded subset of a reflexive Banach space compact in the weak topology?Is every Banach space densely embedded in a Hilbert space?Is a reflexive space necessarily an L-embedded space?Reflexive closure of Banach spaceDoes the norm $mathcalH := | w(x,y)log^+ (w(x,y)) |_L^1$ define a reflexive Banach space?

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Can every Banach space be continuously embedded in a reflexive Banach space? As for example for $ L^infty(Omega)$ and $L^2(Omega)$, where $ Omegasubset R^n$ is a bounded open set.

functional-analysis banach-spaces reflexive-space

share|cite|improve this question

asked Mar 12 at 10:29

RabatRabat

505

$endgroup$

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  The second example you ask about is a Hilbert space, and is therefore a reflexive Banach space.
  $endgroup$
  – Keen-ameteur
  Mar 12 at 10:43
  

  
  
  
  


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  I think you can modify Tomek's answer here to show that the answer is "no" (reflexive spaces have equivalent strictly convex renormings).
  $endgroup$
  – David Mitra
  Mar 12 at 12:01
  

  
  
  
  


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  $begingroup$
  What is "embedding"? So, for example, is the set-theoretic inclusion $L^infty[0,1] to L^2[0,1]$ a "continuous embedding"? You do not require the image to be closed?
  $endgroup$
  – GEdgar
  Mar 12 at 12:49
  

  
  
  
  


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  $begingroup$
  By the continuous embedding, I mean that : $|y|_L^2(Omega)le C |y|_L^infty(Omega), forall yin L^infty(Omega) $ for some constant $C>0,$ without any constraint on the image.
  $endgroup$
  – Rabat
  Mar 12 at 14:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have seen the Tomek's answer. But the strict convexity property is not preserved by equivalence of norm.
  $endgroup$
  – Rabat
  Mar 12 at 14:49
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

Can every Banach space be continuously embedded in a reflexive Banach space? As for example for $ L^infty(Omega)$ and $L^2(Omega)$, where $ Omegasubset R^n$ is a bounded open set.

functional-analysis banach-spaces reflexive-space

share|cite|improve this question

asked Mar 12 at 10:29

RabatRabat

505

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The second example you ask about is a Hilbert space, and is therefore a reflexive Banach space.
  $endgroup$
  – Keen-ameteur
  Mar 12 at 10:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think you can modify Tomek's answer here to show that the answer is "no" (reflexive spaces have equivalent strictly convex renormings).
  $endgroup$
  – David Mitra
  Mar 12 at 12:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is "embedding"? So, for example, is the set-theoretic inclusion $L^infty[0,1] to L^2[0,1]$ a "continuous embedding"? You do not require the image to be closed?
  $endgroup$
  – GEdgar
  Mar 12 at 12:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By the continuous embedding, I mean that : $|y|_L^2(Omega)le C |y|_L^infty(Omega), forall yin L^infty(Omega) $ for some constant $C>0,$ without any constraint on the image.
  $endgroup$
  – Rabat
  Mar 12 at 14:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have seen the Tomek's answer. But the strict convexity property is not preserved by equivalence of norm.
  $endgroup$
  – Rabat
  Mar 12 at 14:49
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Can every Banach space be continuously embedded in a reflexive Banach space? As for example for $ L^infty(Omega)$ and $L^2(Omega)$, where $ Omegasubset R^n$ is a bounded open set.

functional-analysis banach-spaces reflexive-space

share|cite|improve this question

asked Mar 12 at 10:29

RabatRabat

505

$endgroup$

share|cite|improve this question

asked Mar 12 at 10:29

RabatRabat

505

share|cite|improve this question

asked Mar 12 at 10:29

RabatRabat

505

asked Mar 12 at 10:29

RabatRabat

505

asked Mar 12 at 10:29

RabatRabat

505