Definition 4

Tag 02GJ, permanent. Cite it as [Bourbaki, Tag 02GJ].

Théories spectrales, chapter I, § 7. Spectre des endomorphismes des espaces de Banach

Let $E_1$ and $E_2$ be complex Hilbert spaces and $u\in \mathscr{L}(E_1; E_2)$ . The pair $(j,|u|)$ , where $j$ is the unique partially isometric mapping of $E_1$ into $E_2$ such that $u=j|u|$ and Ker( $j$ ) $=$ Ker( $u$ ), is called the polar decomposition of $u$ .

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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