## Abstract and Applied Analysis

### The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces

#### Abstract

We show a necessary and sufficient condition for the existence of metric projection on a class of half-space ${K}_{{x}_{0}^{\ast},c}=\{x\in X:{x}^{\ast}(x)\le c\}$ in Banach space. Two representations of metric projections ${P}_{{K}_{{x}_{0}^{\ast},c}}$ and ${P}_{{K}_{{x}_{0},c}}$ are given, respectively, where ${K}_{{x}_{0},c}$ stands for dual half-space of ${K}_{{x}_{0}^{\ast},c}$ in dual space ${X}^{\ast}$. By these representations, a series of continuity results of the metric projections ${P}_{{K}_{{x}_{0}^{\ast},c}}$ and ${P}_{{K}_{{x}_{0},c}}$ are given. We also provide the characterization that a metric projection is a linear bounded operator.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 908676, 5 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273166

Digital Object Identifier
doi:10.1155/2014/908676

Mathematical Reviews number (MathSciNet)
MR3173296

Zentralblatt MATH identifier
07023289

#### Citation

Zhang, Zihou; Liu, Chunyan. The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces. Abstr. Appl. Anal. 2014 (2014), Article ID 908676, 5 pages. doi:10.1155/2014/908676. https://projecteuclid.org/euclid.aaa/1412273166

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