Assume that H is an operator valid on some hilbert space, N, with nondegenerate eigenvalues, E_n, whose corresponding eigenbasis is {g_n}. If f is a vector that lies on N, then f can be expanded as a linear combination of the eigenbasis of H. Now mathematically, we can say that
H|f> = sum(n, E_n*c_n*|g_n>

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This implies that H changes the the vector, but it is still expanded in the entire eigenbasis. However, if we consider this problem from a physical perspective, |f> collapses into one of the eigenstates it is expressed in, or
H|f> ---> E_n*c_n|g_n>
where there is no summation. Which view is correct?