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 dought 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?

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?