A theorem of Bari on the completeness of orthonormal systems

Yosihiro Urata
1963 Proceedings of the Japan Academy  
The purpose of the present note is to give an another proof of the following THEOREM. If {gn} is a complete orthonormal system of a Hilbert space, and if {n} is an another orthonormal system such as then [} is complete too. The theorem is established by Nina Bari in 1941, according to her obituary note." K. Iski, in a note [2 published in these Proceedings, summarized several extensions of her theorem due to several authors. Recently, G. Birkhoff and G.-C. Rota [1 reproduced the theorem in a connection with the Sturm-Liouville expansions.
doi:10.3792/pja/1195523111 fatcat:adiacbibtvdbbbov7isl4f65qa