# 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.

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Systems are completely described by the observable properties they have (or the degrees of freedom available for measurement or chosen to be described), properties which may not have simultaneously-defined values. This is because observing some aspects of a system may destroy/"undefine" some of the previous properties, precisely those which were incompatible with the new ones. For example, composing Stern-Gerlach apparatus in succession in different axis, allows you to measure the spin components $S_z$ and $S_x$, but after the second filter, a new measurement of the first will be random again, i.e. the observable property "having defined spin in direction __" is not simultaneously defined for independent directions: measuring one makes the system lose its definite value in the other. Another example is the position and linear momentum, which were the commuting degrees of freedom of classical phase space. Quantum mechanical experiments, very related to the originally called "wave-particle duality", determined that position and momentum wave functions (measuring the probability amplitude distributions of possible measurements) were related by a Fourier transformation of each other, thus implying that their commutator was $[\hat x,\hat p_x]=i\hbar$, so they became noncommuting variables better studied under operator algebras.
SUMMARY: Therefore, observables are given by algebras of self-adjoint operators because these are those having real spectra corresponding to the possible real empirical values. Maximal commuting subsets of operators are simultaneously diagonalizable, i.e. they represent compatible observables that can be measured at the same time, so the state of the system at every observation is characterized by such a list of eigenvalues. This defines a Hilbert space by linear superposition upon which the operators act: an observable has a definite value if and only if the state is in one of its eigenvectors. Given an eigenstate in a chosen basis, unitary evolution in Schrödinger's picture moves the vector in the Hilbert space so its projected component to every other possible basis vector changes, where the squared modulus of those complex components of the present vector state over any other vector give the probability of observing the second set of eigenvalues in our next measurements given the first set in our initial measurement. Since these components are given in terms of scalar products $(|\chi\rangle,\, |\psi\rangle)$, by Riesz theorem one can use linear functionals $\langle\chi|$ to act on vectors and give the desired transition amplitudes; since these functionals form also a (dual) linear space, one can think of them as possible final states in our computations. Physical states however are just empirical list of values for every observed degree of freedom of the system, very careful metaphysical considerations must be taken into account before giving meaning to intermediate, not observed superposition states: they statistically correlate observed states at different times but the system cannot be said to have a superposition of its properties. As Dirac claimed: Heisenberg's picture is the [empirical] right one (no one ever observes superpositions!).