Proposition 3

Tag 035Q, permanent. Cite it as [Bourbaki, Tag 035Q].

Théories spectrales, chapter IV, § 5. Opérateurs partiels normaux et théorème spectral

Let $f$ be a universally measurable function defined on the spectrum of $u$ . The set $D_f$ is a dense vector subspace of E. The mapping $y\mapsto$ ev $_y(f)$ is a normal partial operator on E whose domain is $D_f$ , and which is denoted by $f(u)$ .

For every $x\in E$ and every $y\in D_f$ , one has $f\in \mathscr{L}^1$ (Sp( $u$ ) $, \mu_{x,y}$ ) and

\langle x|f(u)y\rangle =\int f \mu_{x,y} \tag{6}

Sp( $u$ )

We shall prove the following more precise statement:

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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