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How do I show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive without using any deep functional analysis theorems?

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Could you be more precise when you say "deep functional analysis theorems"? Perhaps you could state (the names of) result that you are not willing to use. –  matt Nov 7 '12 at 5:01
    
Also, you may find math.stackexchange.com/questions/152343/… helpful. –  matt Nov 7 '12 at 5:02
    
I don't want to use Banach-Alaoglu-Bourbaki theorem, anything related to the Baire Category Theorem, or the Hanh-Banach Theorem. –  Parakee Nov 7 '12 at 12:05
    
@Parkee then you should stop doing mathematics. I'm really hate question like "Prove everything without using anything." Well, one can do that, but this "proof" will repeat standard arguments of Hahn-Banach or Banach-Alaoglu or whatever else, and what is more it will be very long. –  no identity Nov 7 '12 at 17:00
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@Parakee So what has developed your book? –  no identity Nov 7 '12 at 22:23

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