Proposition 8

— Let E be a hilbertian space, M a closed vector subspace of E, I a non-empty ordered directed set and $(M_i)_{i \in I}$ a family of closed vector subspaces of E. We assume that either the mapping $i \mapsto M_i$ is increasing and that M is the closure of $\bigcup_{i \in I} M_i$ or that the mapping $i \mapsto M_i$ is decreasing and that $M = \bigcap_{i \in I} M_i$. Then $p_M(x) = \lim_{i \in I} p_{M_i}(x)$ for all $x \in E$.

Prop. 8 follows immediately from props. 6 (V, p. 11) and 7 (V, p. 12).