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

Abstract

Let $H$ be a real Hilbert space and $C$ a nonempty closed and convex subset of $H$. Let $P_C:H\to C$ denote the (standard) metric projection operator. In this paper, we study the Gâteaux directional differentiability of $P_C$ and investigate some of its properties. The Gâteaux 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 $L^2([-\ddot{I}€,\ddot{I}€])$ with the trigonometric orthonormal basis.

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