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Question about Kadets Klee norm

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$begingroup$

I'm reading a book that claims the following: If $C$ is a closed bounded convex subset of the sphere in a space with the Kadets Klee property, then $C$ is weakly compact.

I'm not sure why this is true. If $(x_n)_n$ is any sequence in $C$. If it has a weakly converging subsequence converging to $x$, then $xin C$ since $C$ is also weakly closed. Then $||x_n_k||to ||x||$ since the sequence and $x$ are in the sphere, and this would imply norm convergence. But I don't see why $(x_n)_n$ has a weakly converging subsequence to begin with.

functional-analysis

share|cite|improve this question

asked Mar 27 at 16:43

user124910user124910

1,262817

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what is Kadets Klee property?
  $endgroup$
  – supinf
  Mar 27 at 16:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you converge weakly and the norms converge, then the sequence converges in norm
  $endgroup$
  – user124910
  Mar 27 at 19:03
  

  
  
  
  

add a comment | 

1

1

$begingroup$

I'm reading a book that claims the following: If $C$ is a closed bounded convex subset of the sphere in a space with the Kadets Klee property, then $C$ is weakly compact.

I'm not sure why this is true. If $(x_n)_n$ is any sequence in $C$. If it has a weakly converging subsequence converging to $x$, then $xin C$ since $C$ is also weakly closed. Then $||x_n_k||to ||x||$ since the sequence and $x$ are in the sphere, and this would imply norm convergence. But I don't see why $(x_n)_n$ has a weakly converging subsequence to begin with.

functional-analysis

share|cite|improve this question

asked Mar 27 at 16:43

user124910user124910

1,262817

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  what is Kadets Klee property?
  $endgroup$
  – supinf
  Mar 27 at 16:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you converge weakly and the norms converge, then the sequence converges in norm
  $endgroup$
  – user124910
  Mar 27 at 19:03
  

  
  
  
  

add a comment | 

$begingroup$

I'm reading a book that claims the following: If $C$ is a closed bounded convex subset of the sphere in a space with the Kadets Klee property, then $C$ is weakly compact.

I'm not sure why this is true. If $(x_n)_n$ is any sequence in $C$. If it has a weakly converging subsequence converging to $x$, then $xin C$ since $C$ is also weakly closed. Then $||x_n_k||to ||x||$ since the sequence and $x$ are in the sphere, and this would imply norm convergence. But I don't see why $(x_n)_n$ has a weakly converging subsequence to begin with.

functional-analysis

share|cite|improve this question

asked Mar 27 at 16:43

user124910user124910

1,262817

$endgroup$

share|cite|improve this question

asked Mar 27 at 16:43

user124910user124910

1,262817

share|cite|improve this question

asked Mar 27 at 16:43

user124910user124910

1,262817

asked Mar 27 at 16:43

user124910user124910

1,262817

asked Mar 27 at 16:43

user124910user124910

1,262817