In David Eisenbud and Joe Harris's book "The Geometry of Schemes", they say some method in algebraic geometry is analogous to the use of distributions in analysis.
In trying to construct a certain scheme, it is often easy to construct the functor that would be the functor of points of that scheme, if the scheme existed; the construction problem is then reduced to the problem of proving that the functor is representable and the use of Yoneda's Lemma (VI-1). The process is exactly analogous to the use of distributions in analysis: there, when trying to prove the existence of a nice function solving a given differential equation, one first proves the existence of a solution that is a distribution, and then is left with the (possibly more tractable) regularity problem of proving that the distribution is represented by integration against a function
I am interested in this process they describe. I know little about differential equations, so I don't know how it work. Are there any books or papers giving this process with easy examples?