Weak Tchebycheff systems (Q1325771)

The author proves the following result: Let \(\{u)i\}_{i\in \mathbb{N}}\) be an infinite integral Chebyshev system on \([a,b]\) and let \(M\) be the closed linear subspace of \(C[a,b]\) generated by \(\{u_ i\}_{i\in \mathbb{N}}\). Then the following are equivalent: (i) \(M\) has the property that for each continuous function \(f\) outside \(M\), there is no best approximation \(\widetilde{f}\) of \(f\) in \(M\); (ii) There is a positive integer \(k_ 0\) such that each system \(\{u_ i\}_{i=1,\dots, k}\) is a Chebyshev system for \(k\geq k_ 0\).