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My question may sound weird and I have no deep motivation behind it other than curiosity.

As is well-known, quasi-reflexive spaces have the Radon-Nikodym property hence their balls have lots of extreme points (they even have the so-called Krein-Milman property). However, can one give me an example of a quasi-reflexive space which is not isometric to a dual space? Of course, every quasireflexive space is isomorphic to a dual space.

I suspect that a clever renorming of the James space should do the job.

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Every non-reflexive Banach space can be equivalently renormed so as not to be isometrically isomorphic to a dual space. $$ $$ Davis, William J.; Johnson, William B. A renorming of nonreflexive Banach spaces. Proc. Amer. Math. Soc. 37 (1973), 486–488.

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    $\begingroup$ @Prawem_Kaduka, well, he had the paper ready since 1973! :-) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Nov 15, 2013 at 19:21

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