We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,\alpha}$. Roughly speaking, under appropriate norm, we could capture the topological information we want. But a question (maybe too vague), how can we know what kind of norm we want in PDE?How can we invent the norm we want? I am just asking for a general idea.



Discovery of norm in PDE
One striking example, though perhaps considered "antique" by now, is Levi's 1906 use of a (true/correct!) minimum principle in Hilbert spaces, and forming what is now called $W^{1,2}$, to prove a (true/correct!) version of "Dirichlet's principle" (which, in effect, had been asserted for certain concrete Banach spaces, for which it did not actually hold...). Also Frobenius in 1907. As it happened, these *true* theorems, replacing arguably-more-natural Banach spaces by Hilbert spaces, sufficed to recover (=give correct proofs of) most of the classical *applications* of "Dirichlet's principle".


Subsequently, Sobolev's work (and Schwartz') vastly expanded this... not only in the Hilbert-space context. For my own applications, the Hilbert-space context is sufficient and extremely useful.


