Hilbert space

The Hilbert space is an essential ingredient in most formalisms of quantum mechanics.
Hence,  any hope of a proper grasp of the counter-intuitive nature of the quantum phenomena must necessarily pass by a good understanding of the concept of the Hilbert space and related notions.
For  one thing the state of the physical system is an element of the Hilbert space. And one should recall that the state of the system encodes everything there is to know about the physical system.
But what is exactly the Hilbert space of quantum mechanics?
The answer in three points.

The Hilbert space is a vector space similar to the three-dimensional physical space we find ourselves inhabiting. Therefore physical states are points inside this vector space in the same way that three-dimensional locations are points inside the physical space.
A point in the Hilbert space is determined by a state vector similarly to the fact that physical locations are determined by position vectors.
The physical states are really synonymous  with the state vectors.
However, there are still two main differences between physical spaces and Hilbert spaces.
Firstly, a Hilbert space is not necessarily three-dimensional but it could have any number of dimensions or even be infinite dimensional. In fact, its basis does not even need to be labelled by a discrete number but the label may as well form a continuum.
Secondly, a Hilbert space is really a complex vector space. Thus, the components of a given state vector in a given basis are given by complex numbers (called wave functions).
This is the most drastic difference between real physical spaces and complex Hilbert spaces.
Indeed, this is the reason why wave functions (which are the complex components of a state vector in a given basis) only compute probability amplitudes. The probability itself is obtained by taking the square of the modulus of the wave function (Born's rule).

The Hilbert space is a complex vector space which is also a metric space.
In other words, we can define norm and distance functions  starting from the notion of the inner or scalar product defined between the state vectors of the Hilbert space. Thus, lengths (norms) of state vectors and distances between state vectors are well defined concept in the Hilbert space.

The Hilbert space is therefore a complex linear vector space which is a metric space.
But the Hilbert space is more than just that.
A Hilbert space is also a complete space (called a Cauchy space).
Intuitively, this means that the Hilbert space does not contain any holes.
From a mathematical point of view this property of completeness means that every series of state vectors which is absolutely convergent  must necessarily converge to some state vector in the Hilbert space. Here, absolute convergence of vectors is nothing  but absolute convergence of their norms, i.e. the sum of their lengths does not diverge.

In summary, the Hilbert space is a complex vector space which is a metric space as well as a complete space.
As a consequence the Hilbert space is an example of the so-called Banach spaces which are spaces in which the whole mathematical structure can be based on the norm and not on the inner product.

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