How to interpret the infinity of Hilbert Space?

[Archeon]

This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite:
\begin{equation*}
\psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty} \hat{c}_k \hat{u}_k(x)
\end{equation*}
So what are the bases u(x)? Are they just other wavefunctions that build a new wavefunction via superposition? And if so, how does this justify the infinite dimensions of the Hilbert space and why exactly is an infinite number of sub-wavefunctions necessary?

Also apologies if I posted this in the wrong subforum, not really sure what this question classifies as.

Thanks in advance.

 

[PeroK]

This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite:
\begin{equation*}
\psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty} \hat{c}_k \hat{u}_k(x)
\end{equation*}
So what are the bases u(x)? Are they just other wavefunctions that build a new wavefunction via superposition? And if so, how does this justify the infinite dimensions of the Hilbert space and why exactly is an infinite number of sub-wavefunctions necessary?

Also apologies if I posted this in the wrong subforum, not really sure what this question classifies as.

Thanks in advance.

Even if you limited yourself to, say, all polynomial functions, then you have an infinite number of basis functions: $1, x, x^2, \dots$