Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{\*\*}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.


Are there examples where $\iota$ fails to be an isomorphism but $V$ and $V^{\*\*}$ are nevertheless isomorphic?


Can one find an example where $V$ is a Banach space and the isomorphism is actually an isometry?



Are there non-reflexive vector spaces isomorphic to their bi-dual?
Yes, the James space.


This is a good question, and R. C. James is rightly praised for this example.



> 
> MR0044024 (13,356d)  
> 
>  James, Robert C.  
> 
>  A non-reflexive Banach space isometric with its second conjugate space.  
> 
>  Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177. 
> 
> 
> 


