# Quasi-reflexive spaces which are not isometric to dual spaces

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.

• Feb 26, 2017 at 12:29

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.