Can subspaces be used to determine probabilities in quantum mechanics?

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This discussion confirms that subspaces can indeed be used to determine probabilities in quantum mechanics. When dealing with an observable that has multiple eigenstates, one can normalize a subset of these eigenstates to 1, thereby obtaining distinct probabilities for that subset. This process is valid for any arbitrary subset of the original eigenstates, which are considered as a subspace of the original Hilbert space. The normalization of probabilities is essential for accurate measurement outcomes in quantum mechanics.

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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?
 
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