The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

Let C be a closed bounded convex subset of a real Banach space X with 0 as its interior and pC the Minkowski functional generated by the set C. For a nonempty set G in X and x ∈ X, g0 ∈ G is called the generalized best approximation to x from G if pC(g0 − x) ≤ pC(g − x) for all g ∈ G. In this paper, we will give a distance formula under pC from a point to a closed hyperplane H(x∗, α) in X determined by a nonzero continuous linear functional x∗ in X and a real number α, a representation of the generalized metric projection onto H(x∗, α), and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.