Summary: In this paper the author studies the existence and the characterization of the \(n\) dimensional Chebyshev subspaces of \(L_\infty[a,b]\), \(B[a,b]\) and some other spaces of discontinuous functions. In the case when the space admits an \(n\) dimensional Chebyshev subspace, the author develops a complete characterization for those \(n\) dimensional Chebyshev subspaces. In the case when the space does not admit an \(n\) dimensional Chebyshev subspaces, the author proves it.