Proposition 10

Let E be a Hilbert space. Then E is an approximation space.

The set A of orthogonal projections of finite rank in E is equicontinuous. Let $n\geqslant 1$ be an integer and $(x_1, . . . , x_n)$ a family of elements of E. Let V denote the vector subspace generated by the $x_i$ and $p_V$ the orthogonal projection with image V. We have $p_V(x_i) =x_i$ for $1\leqslant i\leqslant n$. It follows that $1_E$ is adherent to A for the topology of simple convergence, and hence that E is an approximation space (remark 3).

Other examples of approximation spaces will be given in No.$^o7$ of III, p. 20.