Dtjytyk

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3

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So I am trying to learn some functional analsysis, but the weak star stuff really confuses me. I came across this one and am completely lost. Let $X$ and $Y$ be Banach spaces, and let $F:X^*rightarrow Y^*$ be a bounded linear operator. Show there is a bounded linear operator $G:Yrightarrow X$ with $G^*=F$ if and only if $F:(X^*,wk*)rightarrow(Y^*,wk*)$ is continuous. Thanks for any help you can offer.

functional-analysis

share|cite|improve this question

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it by any chance assumed that the Banach spaces are reflexive?
  $endgroup$
  – pitariver
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Don't need reflexivity. The question is correct.
  $endgroup$
  – Idonknow
  15 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  One direction is easy. Another direction is tricky.
  $endgroup$
  – Idonknow
  15 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

So I am trying to learn some functional analsysis, but the weak star stuff really confuses me. I came across this one and am completely lost. Let $X$ and $Y$ be Banach spaces, and let $F:X^*rightarrow Y^*$ be a bounded linear operator. Show there is a bounded linear operator $G:Yrightarrow X$ with $G^*=F$ if and only if $F:(X^*,wk*)rightarrow(Y^*,wk*)$ is continuous. Thanks for any help you can offer.

functional-analysis

share|cite|improve this question

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it by any chance assumed that the Banach spaces are reflexive?
  $endgroup$
  – pitariver
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Don't need reflexivity. The question is correct.
  $endgroup$
  – Idonknow
  15 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  One direction is easy. Another direction is tricky.
  $endgroup$
  – Idonknow
  15 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

So I am trying to learn some functional analsysis, but the weak star stuff really confuses me. I came across this one and am completely lost. Let $X$ and $Y$ be Banach spaces, and let $F:X^*rightarrow Y^*$ be a bounded linear operator. Show there is a bounded linear operator $G:Yrightarrow X$ with $G^*=F$ if and only if $F:(X^*,wk*)rightarrow(Y^*,wk*)$ is continuous. Thanks for any help you can offer.

functional-analysis

share|cite|improve this question

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

$endgroup$

share|cite|improve this question

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

share|cite|improve this question

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

edited yesterday

pitariver

454113

edited yesterday

pitariver

454113

asked 2 days ago

L. CastaniaL. Castania

161

asked 2 days ago

L. CastaniaL. Castania

161