  Let $H$ be a real Hilbert space and $C$ a nonempty closed and convex subset
of $H$. Let $P_C: H\rightarrow C$ denote the (standard) metric projection
operator. In this paper, we study the G\^ateaux directional differentiability
of $P_C$ and investigate some of its properties. The G\^ateaux directionally
derivatives of $P_C$ are precisely given for the following cases of the
considered subset $C$: 1. closed and convex subsets; 2. closed balls; 3. closed
and convex cones (including proper closed subspaces). For special Hilbert
spaces, we consider directional differentiability of $P_C$ for some Hilbert
spaces with orthonormal bases and the real Hilbert space $L^2([-\pi,\pi])$ with
the trigonometric orthonormal basis.
