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

Abstract

Let $C$ be a closed bounded convex subset of a real Banach space $X$ with $0$ as its interior and $p_C$ the Minkowski functional generated by the set $C$. For a nonempty set $G$ in $X$ and $x\in X$, $g_0\in G$ is called the generalized best approximation to $x$ from $G$ if $p_C(g_0-x)\le p_C(g-x)$ for all $g\in G$. In this paper, we will give a distance formula under $p_C$ from a point to a closed hyperplane $H(x^{\ast },\alpha )$ in $X$ determined by a nonzero continuous linear functional $x^{\ast }$ in $X$ and a real number α, a representation of the generalized metric projection onto $H(x^{\ast },\alpha )$, and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.