Viewing PDEs as maps between function spaces My lecturer remarked the other day that studying PDEs essentially boils down to finding a map between function spaces, and the existance and uniqueness of a solution essentially depends on if this map is surjective and injective, respectively. I didn't get a chance to ask him to elaborate, so let me see if I got the idea right: 
Take a PDE of order $k$ in $\mathbb{R}^n$
$$F(\partial^{k}u,\partial^{k-1}u,...,u,x) = 0$$
with initial conditions 
$$u = \phi ,\text{ on a surface }S \subset \mathbb{R}^n$$
Then essentially, $F$ is a function in some space, say $B_1$ ,and $u$ is a function in a different space $B_2$. So if there is no surjective map from $B_1$ to $B_2$, then there can't be a solution $u$ that satifies the above PDE?  
So hypothetically if we for some spaces $B_1,B_2$ could show that there exist a map $M:B_1 \rightarrow B_2$ that is surjective, then we have proved that a solution exists?
Is this (reasonably) correct?
 A: 
studying PDEs essentially boils down to finding a map between function spaces, and the existance and uniqueness of a solution essentially depends on if this map is surjective and injective, respectively

It really doesn't boil down to that. If you take a brief look of Gilbarg and Trudinger's elliptic PDE book, you will see that most of it is not about maps between  Banach spaces or Hilbert spaces. Or, to pick up another book, Partial Differential Equations by Evans, with emphasis by the author:  

PDE theory is not a branch of functional analysis. Whereas certain classes of equations can profitably be viewed as generating abstract operators between Banach spaces, the insistence  on an overly abstract viewpoint, and consequent ignoring of deep calculus and measure theoretic estimates, is ultimately limiting. 

As for your question: 

Then essentially, $F$ is a function in some space, say $B_1$ ,and $u$ is a function in a different space $B_2$. So if there is no surjective map from $B_1$ to $B_2$, then there can't be a solution $u$ that satifies the above PDE?  

This is false and frankly, makes very little sense. The placement of $F$ and $u$ is wrong. Moreover, the lack of a surjective map does not mean there is no solution for a specific right-hand side.   

So hypothetically if we for some spaces $B_1,B_2$ could show that there exist a map $M:B_1 \rightarrow B_2$ that is surjective, then we have proved that a solution exists?

Also false. The existence of some surjective map $M:B_1\to B_2$ between Banach spaces does not prove anything about the existence of solution to a PDE. Existence would follow if this map was actually an application of the differential operator in the left hand side of the PDE. 
