On existence subspaces in \(C(Q)\) (Q1974363)

The author studies the approximation properties of a Chebyshev subspace \(E\) of a separable space \(C(Q)\) with \(\dim(E)= \operatorname {codim}(E)=+\infty\), in the case where \(E\) or \(C(Q)/E\) is a Lindenstrauss space. One proves a criterion for a subspace \(E\) to be an existence subspace provided that \(C(Q)/E\) is the space of affine continuous functions on a compact Choquet simplex and a criterion for a Chebyshev subspace based on an ``expanded'' representation of a nearest element. This latest criterion implies that if \(Q\) is a countable compact space, then \(C(Q)\) contains no Chebyshev subspaces \(E\) such that \(1\in E\) and \(C(Q)/E\) is a Lindenstrauss space.