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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… – kneidell Sep 20 '11 at 21:18

This is a community wiki answer posted with the goal of getting this question off the unanswered list

This question was crossposted to mathOverflow and received an answer there by GMark:

I'm reproducing the solution here, in case this link is somehow broken:

enter image description here

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