I know that the definition of reflexive module is that the $R$-module $M$ should be isomomorphic to its double dual $M^{**}$ via the canonical map $M\rightarrow M^{**}$.
I'd like to know an example of an $R$-module $M$ such that it's isomorphic to $M^{**}$ but the canonical map $M\rightarrow M^{**}$ is not an isomorphism. Do you have such an example?
(I know that for Banach spaces such example exists, but I don't know it.)
(I put also the tag banach-spaces, maybe it's helpful to know the example for Banach spaces.)