Definition 5
Tag 07XG, permanent. Cite it as [Bourbaki, Tag 07XG].
Topological Vector Spaces, chapter V, § 1. Prehilbertian spaces and hilbertian spaces
A hilbertian space (or Hilbert space) is a prehilbertian space which is Hausdorff and complete. We say that a norm on a vector space E (over K) is hilbertian if it is prehilbertian, and if the normed space E is complete.
If E is a hilbertian space and M a closed vector subspace of E, the prehilbertian space structure induced on M is in fact a hilbertian space structure. In this case we say that M, with the induced structure is a hilbertian subspace of E.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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