Definition 4

Tag 032S, permanent. Cite it as [Bourbaki, Tag 032S].

Théories spectrales, chapter IV, § 4. Opérateurs partiels

Let E and F be normed spaces over K and let $u$ be a partial operator from E into F. For $x$ in dom( $u$ ), one writes

\|x\|_u= (\|x\|^2_E+\|u(x)\|^2_F)^{1/2}

The mapping $x\mapsto \|x\|_u$ resulting is a norm on dom( $u$ ). The normed space thus obtained is denoted by $E_u$ .

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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