A non-reflexive Banach space isometric with A non-reflexive Banach space isometric with its second conjugate space.

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For x = (x1,x2,x3,¼), let ||x|| = sup[å(xpi-xpi+1)2+(xpn+1-xp1)2][1/2] where supremum is over all positive integers n and all finite increasing sequences of at least two positive integers p1,p2,¼,pn+1. Let B be the Banach space of all x for which ||x|| is finite and limn xn = 0. Then B is isometric with B# #, but is isometric under natural mapping with a closed maximal linear subspace of B# #. This example is due to R.C. James (cf. [Jam]).

Ref.

[Jam] R.C.James, A non-reflexive Banach space isometric with its second conjugate space, Proc.of.nat.Acad. of sci., Vol 37, No 3, pp. 174-177, 1951.


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