Just curious, what are the prerequisites for solving these types of questions? Linear algebra, analysis, what else?
For those who are interested, here is one of the exercises from one of my exams you can try to do
This is taken from Jean Saint-Raymond (I quote him so that there is no problem for the origin :s)
Let's denote the Banach space of continuous functions over J with values in with the norm : (uniform convergence norm).
We'll say that if is (that is continuous and differentiable) if f is over and if its derivative has a limit in -1 and 1. We'll also denote the function we then get.
Let's denote the vector subspace of made of the functions over J. We associate to the following norm : .
Let's denote the vector subspace of made of the functions f such that , with the relative norm.
Prove that and are Banach spaces.
Recall the following theorem :
Theorem : Let be a sequence of functions defined over a connected open set of a Banach space E, with values over the Banach space F. We assume that the sequence of differentials converges uniformly to a function g and that there exists such that has a limit in F.
Therefore, the sequence converges uniformly in any ball of to a function, and
Let be the linear mapping . Prove that :
D is bijective
for any and for any
D is an isomorphism of Banach spaces from over (that is to say : the linear mapping is continuous)
We define the function , .
Prove that is a mapping, calculate its differential and prove that (differential of at the point 0) is an isomorphism.
For , let be the open ball, center 0, radius r, of .
Prove that there exists and a function such that for any , is a solution of the differential equation
Haha, good luck
(sidenote : we had a total of 3 hours, and this exercise is worth 1/3 of the marks)