Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System? From my understanding, when we have the Hamiltonian, in principle we can know the eigenstates for our system of interest. Then, we can calculate everything we want. 
In addition, these eigenstates will form a Hilbert space of our quantum system. It seems that having Hamiltonian is enough to specify a quantum System.
However, there are some textbooks which mention we need both Hamiltonian and Hilbert Space to specify a quantum System.
Why do we really need Hilbert space to specify our quantum system?
 A: There is a mathematical reason and a (sort of) physical reason:
Mathematical reason: The Hamiltonian is an operator on a Hilbert space to begin with.  Without knowing a Hilbert space, it doesn't even make sense to speak of an operator on it.  
Physical reason (sort of): What looks like the same Hamiltonian, e.g. the free particle Hamiltonian $H=\frac{P^2}{2m}$, can be meaningfully interpreted in various Hilbert spaces.  For example, the particle could be moving in an unbounded space $\mathbb{R}^n$ or a bounded space like the torus ($\mathbb{R}^n/\mathbb{Z}^n$).  This makes an observable difference: The eigenenergies of the Hamiltonian will be different in the two cases (the spectrum is continuous for $\mathbb{R}^n$ and discrete for the torus).  
(The reason I say the second issue is "sort of" a physical issue is that the mathematical reason given above actually eliminates this problem: To be rigorous you should always specify first a Hilbert space and then a Hamiltonian, and you will never run into ambiguities like whether the spectrum is discrete or continuous.) 
A: Both the Hilbert space and the Hamiltonian are needed, but there exists a duality between the Hilbert space and the Hamiltonian. The more general you make the description of the system and thus the larger the Hilbert space becomes, the simpler the Hamiltonian becomes. 
Consider e.g. the Hamiltonian describing a simple molecule live H2O. This is an extremely complex Hamiltonian, it contains all the interactions between all the electrons in this molecule. However, this Hamiltonian only applies to states where there right number of electrons and nuclei are present. Suppose we  consider a larger Hilbert space and a more general Hamiltonian. E.g. the Hamiltonian of the Standard Model can describe systems that need a larger Hilbert space to be specified. That Hamiltonian is far simpler to specify than the Hamiltonian of an H2O molecule, but you now need more information to specify a state of H2O in the larger Hilbert space that the Standard Model describes.
