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We define, for any group $G$, finite approximations ; with this tool, we givea new presentation of the profinite completion $\hat{\pi} : G \to \hat{G}$ ofan abtract group $G$. We then prove the following theorem : if $k$ is a finiteprime field and if $V$ is a $k$-vector space, then, there is a naturalisomorphism between $\hat{V}$ (for the underlying additive group structure) andthe additive group of the double-dual $V^{**}$. This theorem givescounter-examples concerning the iterated profinite completions of a group.These phenomena don't occur in the topological case.

Profinite completion and double-dual : isomorphisms and counter-examples