On the difficulty of preserving monotonicity via projections and related results (Q2895239)

A subspace \(V\) of a Banach space \(X\) is complemented if there exists a~(bounded) projection mapping \(X\) onto~\(V\). The goal of the present paper is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in \(X = (C[a, b], \|\cdot\|_\infty\)); i. e., finite-dimensional subspaces \(V\subset X\) for which there exists a~projection \(P : X \to V \) such that \(Pf\) is monotone-increasing whenever \(f\)~is. Several corollaries from this consideration are obtained, including a~result describing the difficulty of preserving \(n\)-convexity via a~projection (a~function~\(f\) is said to be \(n\)-convex if, for all choices of \(n + 1\) distinct points \(s_0<s_1<\dotsc<s_n\) in \([a,b]\), the \(n\)th divided difference \(V_n(f;s_i)\) is nonnegative).