Strongly Chebyshev subspaces of matrices (Q1119856)

The author considers the set \(M_ n({\mathbb{R}})\) of real \(n\times n\) matrices as a normed linear space with the spectral norm induced by the Euclidean norm in \({\mathbb{R}}^ n\). Let GL(n;\({\mathbb{R}})\) denote the set of nonsingular matrices in \(M_ n({\mathbb{R}})\). In the first place, the author proves: If \({\mathcal V}\) is a linear subspace of GL(n;\({\mathbb{R}})\), then \({\mathcal V}\) is a Chebyshev subspace of \(M_ n({\mathbb{R}})\), i.e. every vector has a unique best approximation from \({\mathcal V}\).