Characterization of generalized Haar spaces (Q1379571)

Let \(T\) be a connected locally compact metric space and let \(C_0(T,\mathbb R^k)\) be the space of vector-valued continuous functions \(f\) on \(T\) which vanish at infinity. Here \(\mathbb R^k\) is endowed with the Euclidean norm. \textit{S. I. Zukhovitskij} and \textit{S. B. Stechkin} [Dokl. Akad. Nauk. SSSR 106, 773-776 (1956; Zbl 0073.05101)] have characterized finite dimensional subspaces \(G\) of \(C_0(T,\mathbb R^k)\) which are Chebyshev, in terms of generalized Haar subspaces or in terms of strongly unicity of the metric projection \(P_G\). A finite dimensional subspace \(G\) of \(C_0(T,\mathbb R^k)\) is said to be rotation-invariant if \(\{ Qg:g\in G\} = G,\) for any \(K\times K\) orthogonal matrix \(Q\). The metric projection \(P_G\) is said to satisfy the strong unicity condition of order 2 if for every \(f\in C_0(T,\mathbb R^k)\)there exists a constant \(\gamma(f) > 0\) such that \[ |f-g|^2 \geq dist(f, G)^2 + \gamma(f)|g-P_G(f)|^2, \forall g\in G. \] Main result: Let \(G\) be a rotation-invariant finite dimensional subspace of \(C_0(T,\mathbb R^k)\). If \(P_G(f)\) is strongly unique of order 2, whenever \(P_G\) is a singleton, then \(G\) is a Chebyshev subspace of \(C_0(T,\mathbb R^k)\).