I am reading Cohen 'Quantum mechanics' volume 1. In chapter 2, he defines a basis of state space as any set of vectors $|u_i\rangle$ satisfying that for every other vector $| \psi \rangle$ in state space there exists a unique expansion $$ | \psi \rangle = \sum_i c_i | u_i \rangle $$ He does not make any distinction between the case where the state space is finite dimensional and the case where it is infinite dimensional. My doubt is the following: in mathematics, one usually defines the basis of an infinite dimension vector space as a set of vectors such that any other vector can be written as a finite linear combination of the vectors in the basis. Does that mean that for an infinite dimension space state all the $c_i$ coefficients will be $0$ except for a finite amount of them?
In mathematics, one usually defines the basis of an infinite dimension vector space as a set of vectors such that any other vector can be written as a finite linear combination of the vectors in the basis.
This is incorrect - this definition is the notion of a Hamel basis, but the theory of topological vector spaces (in particular of Banach spaces and Hilbert spaces as they appear in quantum mechanics) uses the notion of Schauder bases, where we use the topology of the space to interpret infinite sums in the usual sense of convergence.
For Hilbert spaces and orthonormal bases, this works out to requiring that the coefficients $c_i$ are a square-summable sequence rather than that only finitely many are non-zero.