Is it possible for a Banach Space to be isomorphic to its double dual

[DeadWolfe]

But the cannonical injection to not be an isomorphism?

 

[Pere Callahan]

But the cannonical injection to not be an isomorphism?

In 1950, R. C. James constructed a Banach space which is isometrically isomorphic to its bidual but not reflexive. The natural injection is thus not surjective and therefore no isomorphism.

Reportedly, a proof can be found in Lindenstrauss/Tzafriri "Classical Banach Spaces".