Anywhere, not only in quantum mechanics, where one deals with operators, the space on which these operators operate must be specified; otherwise, one would be dealing with undefined objects. In quantum mechanics, Hilbert space (a complete inner-product space) plays a central role in view of the interpretation associated with wave functions: absolute value of each wave function is interpreted as being a probability distribution function. Probability distributions are by definition non-negative and normalizable (to unity). It is thus that Hilbert space takes a central position in quantum mechanics. I should add that in scattering theory (within the framework of quantum mechanics) one deals with functions that are not normalizable. Here, one deals with fluxes (that is, fluxes of incident and scattered particles), which must be well defined.

Because the evolution operator and observables are (or may be represented by) linear operators acting in a Hilbert space.

The Hilbert space is simply a space of functions, of continuous or discrete variables. All the machinery is but linear algebra applicable to a linear space, which is over C and not R like for the linear space associated to the Euclidean space. The core of quantum mechanics is not in the Hilbert formalism, that through the superposition principle merely expresses the absence of interaction, but in the measuring process, very different from classical wave mechanics, which though is decribed by the very same formalism, including discrete eigenvalues. The Hibert space allows to precisely write down the projection postulate.

Anywhere, not only in quantum mechanics, where one deals with operators, the space on which these operators operate must be specified; otherwise, one would be dealing with undefined objects. In quantum mechanics, Hilbert space (a complete inner-product space) plays a central role in view of the interpretation associated with wave functions: absolute value of each wave function is interpreted as being a probability distribution function. Probability distributions are by definition non-negative and normalizable (to unity). It is thus that Hilbert space takes a central position in quantum mechanics. I should add that in scattering theory (within the framework of quantum mechanics) one deals with functions that are not normalizable. Here, one deals with fluxes (that is, fluxes of incident and scattered particles), which must be well defined.

Using the formalism of Hilbert space calculations in QM becomes

easy. It is also possible to visualize a problem clearly using the

idea of Hilbert space. A common example is harmonic oscillator

problem. Here Schrodinger equation gives infinite number of solutions

with infinite number of eigenvalues. Therefore the oscillator can

reside in any one of those states. In each state the energy eigenvalue

is different. All those (normalized) states span an infinite dimensional

vector space. A physical state can be written as a linear combination

of basis vectors with a definite weight factor for each basis state.

Now a vector space comes with ready-made machinery for

calculations. For example addition, dot product, cross product

and so on. In this "mapping" where a physical problem is

being mapped to a complex vector space, one can use

mathematical machinery of complex vector spaces

for our advantage.

Usefulness of the idea of Hilbert spaces in the context of

quantum mechanics can also be explained using the example

of addition of angular momentum. Consider two electrons

1 and 2 then,

|1/2,1/2> and |1/2,-1/2> spans a 2D Hilbert space.

But because there are two electrons there are two

such Hilbert spaces. After addition is done, one can

represent the product Hilbert space in two ways.

|1/2,1/2>*|1/2,1/2>

|1/2,1/2>*|1/2-1/2>

|1/2-1/2>*|1/2,1/2>

|1/2,-1/2>*|1/2,-1/2>

or one can use the representation

|1,1> |1,0> |1,-1> |0,0>

But these two Hilbert spaces are related.

|1,1> is |1/2,1/2>*|1/2,1/2>

|1,-1> is |1/2,-1/2>*|1/2,-1/2>

|1,0> is |1/2,1/2>*|1/2,-1/2> + |1/2,-1/2>*|1/2,1/2> (symmetric)

|0,0> is |1/2,1/2>*|1/2,-1/2> - |1/2,-1/2>*|1/2,1/2> (antisymmetric)

Idea of forming these basis vectors is that physical

states can be written as a linear combination of basis

vectors. Then one can use all Mathematical machinery

of complex vector spaces.

The notion Hilbert's space is useful because it is adequately describes the mathematics of quantum mechanics. It gives you a mean to think about quantum mechanical systems in general terms engaging your everyday's geometric imagination. You do not need to care about discrete or continuous variables, space or momentum representations etc. Such things ate technical details on that level (although important for eventual calculations).

Quantum Mechanics describes the phenomena at the microscopic level such as atomic and nuclear phenomena. One postulates that the wave function which is generally complex, completely defines the dynamical state of the system being considered, and contrary to classical physics, the dynamical variables of the system such as momentum and position, cannot in general be defined at each instant with infinite precision. If one performs the measurements of a given dynamical variable, the results obtained are statistical in nature and follow some probability law that must be determined by the wave function, the modulus squared of which is a measure for the probability of finding the particle in some region of space. Hence the wave function must be square integrable such that the integral over all space of the modulus squared of the wave function converges according to the Cauchy convergence criterion. The function space that satisfies all these conditions is a Hilbert space which, in addition, is a linear space which allows for the principle of superposition that explains interference phenomena to be applied and in which one can define a scalar product for the purpose of probability interpretations of the results of measurements. That is why the Hilbert space formulation of Quantum Mechanics is so useful to the physicist.

Is there a possibility that take the The Hilbert space corresponding to the server affine space？

QM has the following features: The superposition principle governing the linear Schrodinger equation; The psi function as a mapping in general from R to C (unlike the classical equation of motion, which is a mapping from R to R3); and the Born's probability interpretation, which entails the square-integrability condition.

Because Hilbert space reproduces all these features, it has been "useful" to axiomatize QM. This has been more or less said in all the responses so far. In particular, Dr. Mshelia's answer is very precise and edifying. The original question, however, deserves the addition of a caveat: Hilbert space is needed only in so far as the probability interpretation of the psi function is needed. And the probability interpretation is needed only in so far as we count detector clicks.

Correction to above: (unlike the classical mechanical equation of motion, which yields a mapping from R to R)

Hilbert space brings order in the space of the solutions (set of functions or spin vectors) of quantum mechanical problems. The set of function are chosen to be orthonormal and square integrable. This way you can count and visualize the squares of the wave functions which give the probabilites where the particles are located.

In the end, the main reason for using only Hilbert spaces in quantum theory is technical. What is ultimately observable in experiment is a spectral measure corresponding to an "observable", i.e., a self-adjoint operator in some Hilbert space and a "state," i.e., an element of that Hilbert space. The existence of such spectral measures is guaranteed by the spectral theorems for such operators.  Such theorems are not available for more general cases, e.g., general Banach spaces or normed vector spaces. If they were  available, it would be no problem to use more general spaces as "state spaces" for quantum theory.  For instance, the square of the absolute value of a "wave function," corresponding to a physical state (in the "position representation," where the position operator is "diagonal"),  is an example of a "spectral measure,"  corresponding to an observable (the "position operator") and the state.

Hilbert space has been of pround service to the mathematical physicist. It is an abstract vector space for facilitating the performance of calculations. As such it is not a physically real space. Physicists now suggest that spacetime is not fundamental but emergent; and that quantum mechanics is not complete. Thus we look forward to the next revolution in physics. Perhaps a TOE will lead to a deeper understnding of physical reality and its associated space...

Hilbert space is the only infinite dimensional Banach space such that all of its closed subspaces are complementable.  Propositions in quantum logic correspond to closed subspaces of a Hilbert space, and this logic is not-distributive (hence non-Boolean) due to the existence of incompatible observables (i.e. non-commuting and therefore leading to the uncertainty relations). Naturally, one would likes to have a negation of each proposition, which corresponds to the orthogonal complement of the corresponding subspace.  Thus, each subspace must have its complement.  This is not a problem if the space is finite-dimensional, and arguably one could take any finite dimensional normed linear space over complex numbers together with its dual (they are all isomorphic in finite dimension).  However, we need an infinite-dimensional space if we are going to deal with observables with infinite spectrum. All infinite-dimensional Banach spaces, apart from the Hilbert space, have subspaces without complements.  Thus, if we used a space other than a Hilbert space, then some propositions could not be negated.  Note also that a projector operator onto a subspace exists if and only if the subspace is complemetable.

Another, more abstract reason for using algebra of operators on a Hilbert space comes from the GNS construction and the Gelfand-Naymark theorem for C* algebras (or the Sakai's theorem for von Neumann algebras).

Because of the existence of imaginary numbers in the complex electromagnetic wave system that is created by some degree of capacitance implying a band pass gap, which are explained also by a Fourier transform but implies the uncertainty principle because of the difference between viewing relative to infinity in space, vs infinity in time. While space is infinite time is a mess, while time is infinite space is a mess, which flows on to position and momentum. if it's measured in between it's finite and minimally messy, or on either end it's infinite but rational and irrational.. The square root function to provide irrationality is implied by the electromanetic wave energy magnitudes as they cycle through phases but in 2D dimensions (pythagorean magnitude of the 2D vector necessitates square-root function), thus Hilbert Spaces are square summable and incorporate the Axiom of Choice necessary for the Banach-Tarski situation but operate on the position and momentum uncertainty between time and space implied by the applicability of the Fourier Transform (a conversion between infinite time and infinite space or "Complex Frequencies") to Electromagnetic waves.

Thanks you E.D. Mshelia @adenHandasyde very much for this special view.

Roman V. Belavkin > Propositions in quantum logic correspond to closed subspaces of a Hilbert space, and this logic is not-distributive (hence non-Boolean) due to the existence of incompatible observables (i.e. non-commuting and therefore leading to the uncertainty relations).

I agree very much with this idea, just adding a few more thoughts. Before asking "why Hilbert space?" we should ask "why quantum mechanics?" And to me, the most satisfactory answer was given by quantum logic. I recommend everyone to find and read this old paper:

Article The Logic of Quantum Mechanics

The main idea is that the true logic of experimental propositions should be different from the familiar Boolean logic used in everyday life. In particular, the distributive law should be replaced by a different axiom.

One may protest that usual rules of logic must be untouchable and not a subject to revision. This may be true when we are dealing with classical objects like cars, triangles or natural numbers. They have well defined sharp properties obeying familiar logical relationships. However, when we are dealing with microscopic particles, like electrons, whose observables cannot be measured simultaneously and whose properties exist only probabilistically, we cannot be so sure that classical logic works.

So, Birkhoff and von Neumann declared that the only difference between classical and quantum mechanics is in one single axiom of logic. If you replace the distributive law by the orthomodular axiom, then all other features of the quantum formalism follow automatically. The full proof was given a bit later by Piron:

Book Foundations of quantum physics

In particular, Piron's theorem established that quantum experimental (logical) propositions can be identified with subspaces in the Hilbert space.

Eugene.