Theorem 1

Let E be a prehilbertian space, and H a non-empty convex subset of E such that H is a Hausdorff and complete uniform subspace of E. For every $x \in E$, there exists a unique point $p_H(x)$ in H such that $\|x - p_H(x)\| = \inf_{y \in H} \|x - y\|$. The element $p_H(x)$ of H is also the unique element a of H satisfying the relation

for all $y \in H$.

!Diagram showing a point x, a point y, and a line segment labeled a + λ(y − a)

Fig. 2.

Put $d = \inf_{y \in H} \|x - y\|$, and for every integer $n > 0$, let $H_n$ be the set of points $y$ of H such that $\|x - y\| \leq d + n^{-1}$. The set $H_n$ is closed in H, is convex and non-empty, and its diameter is bounded by $\sqrt{12 \frac{d}{n}}$ for all large enough $n$, by prop. 5. The sequence $(H_n)_{n \geq 1}$ being decreasing, and the set H being Hausdorff and complete it follows that the base of the Cauchy filter $(H_n)_{n \geq 1}$ converges to a point $p_H(x)$ of H; we have $\{ p_H(x) \} = \bigcap_{n \geq 1} H_n$, hence $p_H(x)$ is the unique point $a$ of H such that $\|x - a\| = d$.

1 We recall (GT, VIII, § 1, No. 1) that $\Re(z)$ denotes the real part of the complex number z; we have $\Re(z) = z$ if z is real.

Let $y \in \mathbf{H}$; since $\mathbf{H}$ is convex, the point $z(\lambda) = p_{\mathbf{H}}(x) + \lambda(y - p_{\mathbf{H}}(x))$ of E belongs to $\mathbf{H}$ for every real number $\lambda$ such that $0 < \lambda < 1$. Hence we have

which gives Conversely, let $a$ be a point of $\mathbf{H}$ such that $\mathscr{R} \langle x - a|y - a \rangle \leq 0$ for all $y \in \mathbf{H}$. For every $y \in \mathbf{H}$, we have and so $\| x - a \| = d$ and finally that $a = p_{\mathbf{H}}(x)$ follows from the first part of the proof. Q.E.D.

In what follows the mapping $p_{\mathbf{H}}$ of E in $\mathbf{H}$ will be called the projection from E onto $\mathbf{H}$. We remark that $p_{\mathbf{H}}(x) = x$ for all $x \in \mathbf{H}$.

The first part of th. 1 is valid under more general hypotheses on the space E (V, p. 67, exerc. 31).

The proof of th. 1 establishes, among others, the following property :