Linear combination of eigenfunctions In the textbook "Molecular Quantum Mechanics" by Atkins and Friedman [1, p. 11], it says in chapter 1:


  … a general function can be expanded in terms of all the eigenfunctions of an operator, a so-called complete set of functions. 


Can ANY function be written as a linear combination of eigenfunctions of ANY operator? I think not.

What are the criteria here for which functions can be written as linear combinations of which operators?

References


Atkins, P. W.; Friedman, R. Molecular Quantum Mechanics, 5th ed.; Oxford University Press: Oxford ; New York, 2011. ISBN 978-0-19-954142-3.
The complete, formal mathematical answer is a bit complex. It might be better to get that discussion from Math Stack Exchange. 

An answer to the world of Chemistry is more straightforward: Yes, for any physically relevant operator and any relevant function, you always can. That’s a fundamental property of quantum mechanics: (functional) states are superpositions of eigenstates.