I understand that the operators have to commute, and therefore that the measurement of one has no bearing on the measurement of the other. I know that H, L^2, and L_z form a CSCO for the H atom. Basically, I conceptually understand CSCO. Often though, in my class, we will be working on a problem, finding eigenvalues, and then my teacher says "therefore these form a CSCO." How do you know a set is complete? it seems to me that to make claims of completeness one must try to apply every operator you can think of before excluding any from the set... My problem is recognizing the completeness in the math language. At which point after finding eigenvalues do you see that these form a CSCO? what are the specific mathematical "pictures" that should trigger "oh, these form a CSCO"? if asked "do these form a CSCO?", how do i check that? (other than checking commutation relations.) basically, i think my question can be summed up in: what is the mathematical language of CSCO's, especially in context of the H atom or harmonic oscillator?