The Hilbert Space is the space where wavefunction live. But how would I describe it in words? Would it be something like:

The infinite dimensional vector space consisting of all functions of position $\psi(\vec x)$ given the conditions that $\psi(\vec x)$ is a smooth, continuous function.

(I am not saying this is right, it is merely an example). Further more how does $\Psi(\vec x, t)$ fit into this? Would it be appropriate to say that at any $\Psi(\vec x, t)$ itself is not in the Hilbert space, but at any given time $t_0$ then the function $\psi_0 (\vec x)\equiv \Psi(\vec X,t_0)$ is a member of the Hilbert Space? And all operators $\hat Q$ have to act on members in the Hilbert Space and therefore cannot have time-derivatives but can have time dependencies.

The Hilbert Space is the space where wavefunction live. But how would I describe it in words? Would it be something like:

The infinite dimensional vector space consisting of all functions of position $\psi(\vec x)$ given the conditions that $\psi(\vec x)$ is a smooth, continuous function.

(I am not saying this is right, it is merely an example). Further more how does $\Psi(\vec x, t)$ fit into this? Would it be appropriate to say that at any $\Psi(\vec x, t)$ itself is not in the Hilbert space, but at any given time $t_0$ then the function $\psi_0 (\vec x)\equiv \Psi(\vec X,t_0)$ is a member of the Hilbert Space? And all operators $\hat Q$ have to act on members in the Hilbert Space and therefore cannot have time-derivatives but can have time dependencies.