Corollary 1

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

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

Let I be a set directed by a filter $\mathfrak{F}$ and let $(y_{i})_{i \in I}$ be a family of points of $\mathbf{H}$ . Let $x \in \mathbf{E}$ . Suppose that we have

\lim_{i, \mathfrak{F}} \| x - y_{i} \| = \inf_{z \in \mathbf{H}} \| x - z \| .

Then $y_{i}$ tends to $p_{\mathbf{H}}(x)$ with respect to the filter $\mathfrak{F}$ .

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

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