Corollary 2

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

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

For every $x, y$ in $\mathbf{E}$ , we have

\| p_{\mathbf{H}}(x) - p_{\mathbf{H}}(y) \| \leq \| x - y \| .

In particular, the mapping $p_{\mathbf{H}}$ from $\mathbf{E}$ into $\mathbf{H}$ is continuous.

Let $x, y$ be two points of $\mathbf{E}$ . Put $a = p_{\mathbf{H}}(x) - x, b = p_{\mathbf{H}}(y) - p_{\mathbf{H}}(x), c = y - p_{\mathbf{H}}(y)$ . By formula (15) (V, p. 10) we have $\mathscr{R} \langle a|b \rangle \geq 0$ and $\mathscr{R} \langle c|b \rangle \geq 0$ . We also have $a + b + c = y - x$ , which gives,

\| x - y \|^{2} = \| a + b + c \|^{2} = \| b \|^{2} + \| a + c \|^{2} + 2 \mathscr{R} \langle a|b \rangle + 2 \mathscr{R} \langle c|b \rangle

\geq \| b \|^{2} = \| p_{\mathbf{H}}(x) - p_{\mathbf{H}}(y) \|^{2} .

This proves corollary 2.

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

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