Continuity of Metric Projection Operator from C[0, 1] onto Pn with Applications to Mordukhovich Derivatives

TL;DR

This paper proves the continuity of the metric projection from C[0, 1] to polynomial subspaces and explores its implications for Mordukhovich derivatives, including properties and fixed points.

Contribution

It establishes the norm continuity of the metric projection operator onto polynomial subspaces in C[0, 1], enabling analysis of Mordukhovich derivatives.

Findings

Metric projection operator is single-valued and continuous.

Continuity facilitates the study of Mordukhovich derivatives.

Results have applications in fixed-point theory.

Abstract

Let C[0, 1] be the Banach space of all continuous real valued functions on [0, 1]. For an arbitrarily given nonnegative integer n, let Pn denote the set of all polynomials with degree less than or equal to n. Pn is a closed subspace of C[0, 1]. In this paper, we first prove (in details) that the metric projection operator from C[0, 1] to Pn is a single-valued mapping and it is (norm to norm) continuous. Then, we use the continuity of Pn to investigate the Gateaux directional derivatives, some properties and fixed-point properties of the Mordukhovich derivatives of the metric projection operator.