Many expositions of Quantum Mechanics begin by stating axioms of the following type:
"The state of a quantum mechanical system is given by a nonzero vector in a complex vector (or Hilbert space) with an Hermitian Inner Product"
These types of expositions then go on to define observables as self-adjoint linear operators etc.
Landau and Lifshitz build up their edifice in a seemingly different way, beginning with an assertion (or perhaps, for them, a kind of axiom) that probabilities are bilinear on $\Psi$ and $\Psi^{*}$ (I think most of us would now instead say that it was a sesquilinear form rather than conjugate-linear). They then deduce conclusions throughout the exposition. In particular they never explicitly assert that states form a vector space (which then would have a basis).
My difficulty stems from discerning whether, at various points, certain statements are to be taken as "axiomatic" or derived. Specifically:
Landau and Lifshitz, in section 3 of Quantum Mechanics, state that (I will refer to this as statement 1):
"In accordance with the principle of superposition, we can assert that the wave function $\Psi$ must be a linear combination of those eigenfunctions $\Psi_n$ which correspond to the values $f_n$ that can be obtained, with probability different from zero, when a measurement is made on the system and it is in the state considered."
Their previous statement of the superposition principle (in section 2) is as follows:
Suppose that, in a state with wave function $\Psi_1 (q)$, some measurement leads with certainty to a definite result (result 1), while in a state $\Psi_2 (q)$ it leads to result 2. Then it is assumed every linear combination of $\Psi_1 (q)$ and $\Psi_2 (q)$ ... gives a state in which measurement leads to either result 1 or result 2.
In this superposition statement (sect 2) L&L have not explicitly stated the converse; that every state in which a measurement leads to either result 1 or result two can be expressed as a linear combination of the eigenfunctions $\Psi_1 (q)$ and $\Psi_2 (q)$. (And this would not be unique - for example an electron state in which there was equal chance of being z up or z down could potentially be in x up or x down etc.)
The Question I am trying to ask is:
"Is L&L's statement 1 to be taken as axiomatic? Or is it derived?"
I presently see it as probably axiomatic because I cannot yet derive it from previous statements (as explained in the paragraph below), but I am not sure, hence the question. It seems to me to be perhaps equivalent to saying that quantum states must live in a vector space.
I can see why $\Psi$ could be written as a linear combination of those eigenfunctions (from their statement of the principle of superposition), but it also seems to me that that when L&L assert that the wave function $\Psi$ must be a linear combination of those eigenfunctions $\Psi_n$ they may be introducing a new physical principle of perhaps greater depth than simply deriving from the principle of superposition. In particular it seems to be equivalent to saying that these eigenfunctions form the basis for a vector space.
Further clarification
The assertion I am referring to is the "assert" in bold in the first quotation.
