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The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia)

Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over the complex numbers.

Moreover, we know that in order to have a useful representation we need to ensure that the vector representing the quantum state is a *unit vector*.

But in the definition above, they don't precise the norm (or the scalar product) associated with the Hilbert space considered. At first glance I though that the norm was not really important, but I realised yesterday that the norm was *everywhere* chosen to be the Euclidian norm (2-norm).
Even the bra-ket notation seems to be made specifically for the euclidian norm.

**My question:** Why is the Euclidian norm used everywhere? Why not using an other norm? Does the Euclidian norm has useful properties that can be used in quantum mechanics that others don't?

2Actually I just wanted to add a comment but I don't have the reputation for it: note that, as you write in your question - quantum states are rays in the Hilbert space. This means that they are not normalized, but rather that all vectors in the Hilbert space that point in the same direction are equivalent. It is more convenient to work with normalized states but the physics is actually hidden in the overlap of the states with each other. It is for this reason that there is no norm present in the definition of a state. – Omri Har-Shemesh – 2018-07-14T07:09:08.080