I'm trying to learn basic quantum mechanics by myself. As I understand now, any quantum system can be described by an abstract state vector $\left\vert \Psi (t) \right\rangle$ which generally can be considered a function of time. Now this can be expressed as a linear combination of different basis eigenvectors. As Susskind states in its "Quantum mechanics: the theoretical minimum" eigenvectors of an hermitian operator form an orthonormal basis. I understand that examples could be position, momentum or energy operators, which can be used to write the state vector as follows:
$$ \{\left\vert x \right\rangle\} \to \left\vert \Psi(t) \right\rangle = \int \left\langle x \vert \Psi(t) \right\rangle \left\vert x \right\rangle dx = \int \psi(x) \left\vert x \right\rangle dx $$
$$ \{\left\vert p \right\rangle\} \to \left\vert \Psi(t) \right\rangle = \int \left\langle p \vert \Psi(t) \right\rangle \left\vert p \right\rangle dp $$
$$ \{\left\vert E \right\rangle\} \to \left\vert \Psi(t) \right\rangle = \sum_i \left\langle E_i \vert \Psi(t) \right\rangle \left\vert E_i \right\rangle = \sum_i C_i \left\vert E_i \right\rangle$$
as it is shown in this video (minute 1:30). Since it is postulated that $\left\vert \Psi(t) \right\rangle$ belongs to a Hilbert space and can be expanded in terms of an infinite number of eigenvectors which form a basis, the dimensionality of its vector space should be infinite.
On the other hand, there are cases in which the state vector can be expressed as a linear combination of a finite number of eigenvectors, such as when speaking about spin. For example, when analyzing the Stern-Gerlach experiment it is useful to express $\left\vert \Psi(t) \right\rangle$ as:
$$ \left\vert \Psi(t) \right\rangle = \alpha_{\downarrow} \left\vert \downarrow \right\rangle + \alpha_{\uparrow} \left\vert \uparrow \right\rangle $$
The dimension of a vector space is defined to be the number of vectors in one of its basis. Probably I'm missing something about the nature of the state vector, but how is it possible for it to be expanded in terms of infinite and finite sets of eigenvectors? Moreover, even if both infinite, the number of eigenvectors in $\{\left\vert x \right\rangle\}$ and $\{\left\vert p \right\rangle\}$ should be greater than the one in $\{\left\vert E \right\rangle\}$, shouldn't this be same inequality that relates the cardinality of the real and the rational numbers?
Thank you!
P.S.: My guess is that when talking about spin, we are talking about a quantum system itself, that thus requires a state vector to be described, different from the one used to describe a whole particle, for example. Therefore, when talking about a particle with spin, it is as if the two system are merged together...
