Simpler Proof of the Ringrose's Characterization of Compact Operators

Abstract

The aim of this note is to give short, simpler and elementary proof of one characterization of compact operators via orthonormal sequences, which, hopefully will make this fact more accessible to nonspecialists and to a wide audience (especially to students). Beside this, the proof given here shows that this fact holds true for operator acting from Hilbert space to some Banach (not necessarily Hilbert) space.