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\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.