After Newton's success in giving a mathematical account of planetary motion, scientists were emboldened to attempt the description of many physical phenomena using the differential calculus. The ‘infinitesimal’ viewpoint gave a fruitful way of formulating laws governing everyday processes or their idealizations. Wave motion, fluid flow and the conduction of heat were analysed. Their description is harder than that of celestial motions, if only because the world of our experience has such an abundance of detail in comparison with the emptiness of space. Still, there is a miraculous feature of the fabric of the universe to hearten the mathematician: a great diversity of physical processes is well described by second-order linear partial differential equations. Much of functional analysis stems from the study of such equations.

Imagine that you are a pioneer of mathematical physics faced with a new class of equations of the general type Lf = g. Here g is supposed to be a known function of space and time variables, L is a linear differential operator and f is a function which is unknown but required to satisfy some initial or boundary conditions. Your first concern would be to devise a way of finding f. Even if you work out a bag of tricks that does well in practice, think of the difficulty of proving that a particular technique will always succeed for the class of equations under study.