I have been reading David Albert's book "Quantum Mechanics and Experience", and came across the following statement in chapter 2 page 41: "(5) Any vector whatever in a given space will invariably be an eigenvector of some complete Hermitian operator on that space. That....will entail that any quantum state whatever of a given physical system will invariably be associated with some definite value of some measurable property of that system."
I am having some difficulty understanding this statement intuitively. I am particularly puzzled by the use of the words "invariably" and "whatever". It seems to me that it is saying that given an arbitrary vector in a vector space, I can always find some operator (corresponding to a measurable/observable) for which it is an eigenvector. What is an example of such an observable? Can there be (infinitely) many such observables? Does it depend on the chosen vector space?
I would like to understand this intuitively, if possible, with some examples. Of course, a formal proof is also helpful. Please note that I do understand vector spaces and other formalism, but I have not come across anything like this particular statement before or when searching the literature.
Thanks in advance for any clarification.