Spectral theory and separation of variables

For many of the partial differential equations discussed above, it's the separation-of-variables method (consisting of searching for possible solutions of the form : $\left(x,\mathrm{\hspace{0.17em}}y\right)\to u\left(x\right)v\left(y\right),$

and then to form linear combinations of them, possibly infinite in some sense to be specified) which has often made it possible to determine the sought-after solutions.

The theoretical (and also practical for many simple special cases) success of this method rests on the foundations we now set out. In the preceding paragraphs, we saw that...