  If $K$ and $L$ are mutually dual closed convex cones in a Hilbert space with
the metric projections onto them denoted by $P_K$ and $P_L$ respectively, then
the following two assertions are equivalent: (i) $P_K$ is isotone with respect
to the order induced by $K$ (i. e. $v-u\in K$ implies $P_Kv-P_Ku\in K$); (ii)
$P_L$ is subadditive with respect to the order induced by $L$ (i. e.
$P_Lu+P_Lv-P_L(u+v)\in L$ for any $u, v \in \R^n$). This extends the similar
result of A. B. N\'emeth and the author for Euclidean spaces. The extension is
essential because the proof of the result for Euclidean spaces is essentially
finite dimensional and seemingly cannot be extended for Hilbert spaces. The
proof of the result for Hilbert spaces is based on a completely different idea
which uses extended lattice operations.
