I'm self studying functional analysis this summer so I'll be prepared for my next semester, and I wonder how it helps in the study of differential equations.
From my understanding functional analysis is essentially infinite dimensional linear algebra, and since derivatives are linear operators it is fairly obvious that it should be helpfull in some way, but exactly how does it help? Does it allow us to find solutions? Does it help prove existance? Uniqueness, etc? If so, how does it go about proving these statements.