On Nonlinear Idempotents and Metric Projections in Normed Spaces

Himanshu Kumar; Abdullah Bin Abu Baker; Fernanda Botelho

On Nonlinear Idempotents and Metric Projections in Normed Spaces

Functional Analysis 2026-07-16 v1

Authors: Himanshu Kumar , Abdullah Bin Abu Baker , Fernanda Botelho

Abstract

Let $X$ be a normed linear space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$ . In this paper, we introduce some new classes of idempotent maps (linearity is not assumed) on $X$ and study their properties. A well-known example of an idempotent map is the metric projection $P_K$ onto a Chebyshev subset $K$ of $X$ . We provide some necessary conditions for the map $I- P_K$ to be idempotent, where $I$ denotes the identity operator on $X$ .

Keywords

idempotent map metric projection Chebyshev set normed space nonlinear operator

Cite

@article{arxiv.2607.14649,
  title  = {On Nonlinear Idempotents and Metric Projections in Normed Spaces},
  author = {Himanshu Kumar and Abdullah Bin Abu Baker and Fernanda Botelho},
  journal= {arXiv preprint arXiv:2607.14649},
  year   = {2026}
}

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8 pages