Projection Operators

Abstract

Let C be a nonempty closed convex subset of $\mathcal{H}$. The projection $P_{c}x$of a point $x\, \in \, \mathcal{H}$ onto C is characterized by (see Theorem 3.14)

$$P_{c}x \in C \quad {\rm and} \quad (\forall y \in C) \quad \langle y-P_{c}x | x-P_{c}x \rangle \leq 0$$

or, equivalently, by (see Proposition 6.46) $x - {P_{c}x } \in N_c\left( {P_{c}x } \right)$). In this chapter, we investigate further the properties of projectors and provide a variety of examples.