An exercise from my topology & differential calculus exam
For those who are interested, here is one of the exercises from one of my exams you can try to do :D
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 :D
(sidenote : we had a total of 3 hours, and this exercise is worth 1/3 of the marks)