L'espace usuel C ne possède aucun R.U.C-système orthonormé. (The usual space C doesn't have an orthonormal R.U.C-system) (Q1822723)

It is proved that a sequence \((\phi_ n)\) in the usual Banach space C which is orthonormal and fundamental (for example the Legendre system), never is an orthonormal R.U.C-system (i.e. has the additional property that for each f in C with the expansion \(\sum a_ n\phi_ n\) we have convergence of \(\sum \pm a_ n\phi_ n\) in C for almost all choices of signs \(\pm).\) The meaning of this result is to be found in comparison with known results, especially i) there exists a basis \((\phi_ n)\) in C such that for each f in C with the expansion \(\sum a_ n\phi_ n\) we have the convergence of \(\sum \pm a_ n\phi_ n\) in C for almost all choices of signs \(\pm,\) ii) for no basis in C the words ``almost all'' in i) can be replaced by the work ``all'', iii) there exist an orthonormal basis in C (for example the Franklin system).