On the Riesz-Fischer theorem

Are the two forms in which the theorem of the title is usually stated equivalent? We first summarize the three Comptes Rendus notes in which Frédéric Riesz published his results concerning L2, and then, in somewhat more detail, an article from 1910 which has been published only in Hungarian. Riesz deduces the two forms not from each other but both from the Fréchet—Riesz representation theorem. A theorem states that some of Riesz's results hold in the case of an abstract inner product space, and leads to maximal orthonormal systems which are not total. We conclude with a proof due to Ákos Császár which shows that a variant of Riesz's condition implies the Fischer form (i.e., completeness).