The linearity of metric projection operator for subspaces of L p spaces

Abstract

Let Y be a Chebyshev subspace of a Banach space X. Then the single-valued metric projection operator PY: X ∈ Y taking each x ∈ X to the nearest element y ∈ Y is well defined. Let M be an arbitrary set and µ be a σ-finite measure on some σ-algebra Σ of subsets of M. We give a description of Chebyshev subspaces Y ⊂ Lp(M,Σ, µ) with a finite dimension and a finite codimension which the operator PY is linear for.