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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)

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For Banach spaces, one example is James's Construction: See mathoverflow.net/questions/43986/… –  kneidell Sep 20 '11 at 21:18

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