Example 3

Tag 07XD, permanent. Cite it as [Bourbaki, Tag 07XD].

Topological Vector Spaces, chapter V, § 1. Prehilbertian spaces and hilbertian spaces

Let $n \geqslant 0$ be an integer. We define a prehilbertian space structure on the space $K^n$ , by means of the hermitian form

(x, y) \mapsto \sum_{j=1}^n \overline{x_j} y_j

(for $x = (x_1, ..., x_n)$ and $y = (y_1, ..., y_n)$ ). When $K$ is $\mathbf{R}$ , we see that this is just the scalar product of two vectors of $\mathbf{R}^n$ (GT, VI, § 2, No. 2).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

VI, § 2, No. 2 not yet on this site