Corollary 3

Tag 08XN, permanent. Cite it as [Bourbaki, Tag 08XN].

— Let $\mu$ be a positive measure on $X$ , $F$ a real (resp. complex) Hilbert space. On the space $L_F^2$ , the symmetric (resp. Hermitian) form

(\widetilde{f}, \widetilde{g}) \mapsto \int \langle f, g \rangle\, d\mu

defines a Hilbert space structure, for which the norm is equal to $\| \widetilde{f} \|_2$ .

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

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