Can subspaces be used to determine probabilities in quantum mechanics?

[friend]

Suppose we have an observable with a certain number of eigenstates. We would normalize all these possibilities to 1 in order to give each eigenstate an appropriate probability of being measured. Can we then only consider the data of many measurements for only a subset of those eigenstates and normalize that subset to 1 and get different probabilities for considering only that subset of alternatives? Is that subset called a subspace of the original Hilbert space? And can this be done for any arbitrary subset of the original eigenstates?

 

[Demystifier]

Yes, yes and yes.