Example 2

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

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

\* Let X be a Hausdorff topological space and let $\mu$ be a positive measure on X. Let $L^2(X, \mu)$ be the space consisting of equivalence classes, for $\mu$ , of all square $\mu$ -integrable functions on X with values in $\mathbf{C}$ . This is a complex hilbertian space, whose scalar product is given by

\langle f|g \rangle = \int_X \overline{f(x)}\ g(x)\ d\mu(x) .

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

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