Remark
Tag 05SJ, permanent. Cite it as [Bourbaki, Tag 05SJ].
Algebra, chapter II, § 7. Vector spaces
An infinite-dimensional subspace G' of E* is not necessarily the orthogonal of a subspace of E, in other words, if F is the orthogonal of G' in E, the orthogonal F' of F in E* may be distinct from G' (Exercise 20(b)).†
† By giving E and E* suitable topologies and only considering in E and E* subspaces closed with respect to these topologies, it is possible to re-establish a perfect symmetry between the properties of E and E* when E is infinite-dimensional (cf. Topological vector spaces, II, § 6).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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