Proposition 3
Tag 07ZH, permanent. Cite it as [Bourbaki, Tag 07ZH].
Topological Vector Spaces, V, p. 21, § 2. Orthogonal families in a hilbertian space
In a Hausdorff prehilbertian space E, every orthonormal family is topologically independent.
We note that this property immediately follows from the characterization of topologically independent families (IV, p. 1 and II, p. 43, cor. 2), on account of the identification of the dual of E with the completion of E or with the space conjugate to E according as K is equal to R or C (V, p. 17, Remark).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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