## Abstract and Applied Analysis

### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 504076, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449680

Digital Object Identifier
doi:10.1155/2013/504076

Mathematical Reviews number (MathSciNet)
MR3132553

Zentralblatt MATH identifier
1303.46016

#### Citation

Luo, XianFa; Wang, JianYong. The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 504076, 6 pages. doi:10.1155/2013/504076. https://projecteuclid.org/euclid.aaa/1393449680

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