Proposition 5

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

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

— Let $E$ be a prehilbertian space. Let $d$ be a real number $> 0$ , $\delta$ a real number such that $0 \leqslant \delta \leqslant d$ . Let $B$ and $B'$ be subsets of $E$ defined by $\| x \| < d$ ,

\|x\| \leq d + \delta \text{ respectively, and let } A \text{ be a convex set contained in } B' - B. \text{ Then for every pair of points } x, y \text{ of } A, \text{ we have } \|x - y\| \leq \sqrt{12d\delta} \text{ (fig. 1).}

In fact, we have

\frac{1}{2}(x + y) \in A

, hence

\left\| \frac{1}{2}(x + y) \right\| \geq d

; hence from (14) we get the inequality

\left\| \frac{1}{2}(x - y) \right\|^2 = \frac{1}{2}(\|x\|^2 + \|y\|^2) - \left\| \frac{1}{2}(x + y) \right\|^2 \leq (d + \delta)^2 - d^2 \leq 3d\delta

from which the proposition follows.

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

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