quantum mechanics - Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:

1. 
   

   The observables are given by self-adjoint operators on the Hilbert Space.

   
   
   


2. 
   

   Gelfand-Naimark Theorem implies a duality between states and observables

   
   


3. 
   

   What's the significance of spectral decomposition theorem in this context?

   
   


4. 
   

   What do the Hilbert Space itself corresponds to and why are states given as functionals on the Hilbert space.

   
   

I need a real picture of this. I posted in Math.SE but got no answer. So I am posting it here.