$$i\hbar \frac{\partial}{\partial t}\psi(x,t)=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\partial(x,t)+U(x)\psi(x,t)$$
Why is every solution of the above equation either of the form $\phi(x)T(t)$ or a linear combination of functions of the form $\phi(x)T(t)$ i.e. $c_1\phi_1(x)T_1(t)+c_2\phi_2(x)T_2(t)+...$ ?
[This was actually claimed in "Introduction to Quantum Mechanics by David J. Griffiths",Chapter 2]
It isn't, though physicists might pretend that it is. If you add the important qualifications that $\psi(x,t)$ is square-integrable for fixed $t$, and that the Hamiltonian is self-adjoint, you need infinite series in the case of discrete spectrum, and integrals in the case of continuous spectrum.