Say we have an observable that describes a Hilbert space and that observable acts on state kets. Lets take the position observable for example. Then ⟨y|x⟩=δ(y−x). But can the eigenstates of the position observable be individually thought of as delta functions? A|x⟩=x′|x⟩
Is this |x⟩ then individiually a delta function picking x′ out of A? Wouldn't this also imply that we have an infinite number of delta function eigenstates in the observable space?
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