Linearity of metric projections on Chebyshev subspaces in \(L_1\) and \(C\) (Q1280601)

Let \(Y\) be a closed linear subspace of a Banach space \(X\), \(x\in X\) and \(P_Y(x)= \{y\in Y:\| x-y\|= \text{dist}(x,Y)\}\) be the set of elements of \(Y\) nearest to \(x\). \(Y\) is called a Chebyshev subspace if the set \(P_Y(x)\) is a singleton for each \(x\in X\). The mapping \(P_Y: X\to Y\) that takes each \(x\in X\) to the best approximation element \(P_Y(x)\in Y\) is called the metric projection operator. Let \(L_1(M,\Sigma,\mu)\) denote the space of real-valued functions defined on an arbitrary set \(M\) and integrable on \(M\) with respect to a \(\sigma\)-finite measure \(\mu\) defined on a \(\sigma\)-algebra \(\Sigma\) of subsets of \(M\). Let \(C(K)\) denote the space of real-valued continuous functions on a compact Hausdorff space \(K\) equipped with the uniform norm \(\| f\|= \sup\{| f(t):t\in K\}\). In general, the metric projection operator \(P_Y\) is neither linear nor continuous. The aim of the present paper is to describe Chebyshev subspaces \(Y\) in \(L_1(M,\Sigma,\mu)\) and \(C(K)\) such that the operator \(P_Y\) is linear (Theorems 1 and 3). Moreover, a wide class of Chebyshev subspaces in \(L_1(M,\Sigma,\mu)\) for which the operator \(P_Y\) is nonlinear in general has also been indicated in this paper (Theorem 2).