Do you know an example of a space U^\perp\perp that is not a subset of U with following conditions? Can it be from l^2?

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Alexander Alldridge Popular answer

If U is a subspace, then U^{\perp\perp} is its closure (by the Hahn-Banach theorem), so you may take any subspace that is not closed as an example. For instance, if X=\ell^2, then you may take U to be the space of all finitely supported sequences.

Alexander Alldridge