An exercise from my topology & differential calculus exam

Hi !

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 .

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.

**Q.1**

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

**Q.2**

Let be the linear mapping . Prove that :

for any

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)

**Q.3**

We define the function , .

Prove that is a mapping, calculate its differential and prove that (differential of at the point 0) is an isomorphism.

**Q.4**

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

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Haha, good luck :D

(sidenote : we had a total of 3 hours, and this exercise is worth 1/3 of the marks)