Any ket as a sum of eigenkets of a given observable Most quantum mechanics texts include a phrase such as 'any ket can be written as a sum of eigenkets of a given observable'.
I have problems with the generalities of any ket.
Does this literally mean any ket, or does it mean any ket of the same observable? It seems odd that any ket, say an eigenket of position, should be expressible as a sum of eigenkets of a different observable?
Edit: 
Unless of course a general state is dependent on more than one summation of eigenkets of more than one observable, but the measurement of one of those observables does not depend on the other summations. 
 A: Think about it this way: When you measure some observable $O$, you get some measured value. If you measure many identically prepared systems, you may measure different values corresponding to the operator $O$. In the limit where you do this infinitely many times, you are able to recover exactly which values are possible, and with which probability. Basic quantum theory now tells you that the system (before you measured anything) was in a superposition of eigenstates corresponding to the (eigen)values for $O$ which you ended up measuring. 
This means that the system is in the state $$|\psi\rangle =\sum_{i=1}^\infty c_i |o_i\rangle$$ i.e. it can be written as a weighted sum of eigenkets $|o_i\rangle$ of $O$. This basically just expresses that one can always measure the observable $O$, and get some value. In some cases, the sum should be replaced by an integral, or it can be a sum over finitely many eigenstates, but the idea is the same.
A: Look it like this, a ket is a vector of a (fancy) vector space and as such it has different components with different weights. 
Now, going to a more familiar vector space of the $R^3$, you can choose for example the cartesian coordinate system to describe a vector in space, but you can also choose any other set of three vectors (provided they're not colinear) to describe such a vector. In particular you can pick any rotation of the initial cartesian system.
That procedure and possibility in real space, also translates to the Hilbert vector space where kets (and bras) "live". It's nothing else than to say you're expressing your vector in a different basis.
