Directional Differentiability of the Generalized Metric Projection in Hilbert spaces and Hilbertian Bochner spaces

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.