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


1. 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](https://math.stackexchange.com/). 


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. 


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