Finding all solutions of the wave equation using $u(x,t) = X(x)T(t)$

Question

Assuming that function $u(x,t)$ in the wave equation $\partial_t^2u = c^2\nabla^2u$, can be factorized to $u(x,t) = X(x)T(t)$, this yields a viable strategy for solving the wave equation, and explicit solutions can be found for $X$ and $T$, and an infinite number of solutions may be found of the form $u(x,t) = \sum_i a_iX_i(x)T_i(t)$. However, does this space cover all the solutions of the wave equation, or just an infinite number of them?

Answer 1

The solutions to these sort of PDEs generally lie in a Sobolev space. If this space is in addition a separable Hilbert space, countable solution sets of the form $X_i(x)T_i(t)$ which are orthogonal wrt the inner product of the space form a basis for the space, thus every solution to the equation must admit the representation $\sum_ia_iX_i(x)T_i(t)$ for some (typically infinite) set of coefficients $a_i$.