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Is there a Hilbert Space for each energy level of a system? (And, in general, for every point in time?)

I read in some book that if a equation for a problem accepts two different sets of wavefunction solutions (the case in question was the free particle and the sets of solutions in Cartesian and spherical coordinates) then the functions in one of those sets could be expressed by a linear combination of solutionsof the same energyof the other set because of the completenes of the sets of eigenvectors of Hermitian operators.

But a complete set of vectors should span the whole hilbert Space, hence my question.

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# Energy levels and Hilbert Spaces

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