On Nonlinear Idempotents and Metric Projections in Normed Spaces

2026/07/16 by Himanshu Kumar, Abdullah Bin Abu Baker, Fernanda Botelho
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Abstract

Let X be a normed linear space. A map P: X → X is called idempotent if P² = 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.

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