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 fromJean Saint-Raymond(I quote him so that there is no problem for the origin :s)

Let $\displaystyle J=[-1,+1]$.

Let's denote $\displaystyle \mathcal{C}^0(J)$ the Banach space of continuous functions over J with values in $\displaystyle \mathbb{R}$ with the norm : $\displaystyle \|f\|_0=\sup_{t \in J} |f(t)|$ (uniform convergence norm).

We'll say that if $\displaystyle f \in \mathcal{C}^0(J)$ is $\displaystyle C^1$ (that is continuous and differentiable) if f is $\displaystyle C^1$ over $\displaystyle ]-1,+1[$ and if its derivative $\displaystyle f'$ has a limit in -1 and 1. We'll also denote $\displaystyle f'$ the function we then get.

Let's denote $\displaystyle \mathcal{C}^1(J)$ the vector subspace of $\displaystyle \mathcal{C}^0(J)$ made of the $\displaystyle C^1$ functions over J. We associate to $\displaystyle \mathcal{C}^1(J)$ the following norm : $\displaystyle \|f\|_1=\|f\|_0+\|f'\|_0$.

Let's denote $\displaystyle \mathcal{C}_0^1(J)$ the vector subspace of $\displaystyle \mathcal{C}^1(J)$ made of the functions f such that $\displaystyle f(0)=0$, with the relative norm.

Q.1

Prove that $\displaystyle \mathcal{C}^1(J)$ and $\displaystyle \mathcal{C}_0^1(J)$ are Banach spaces.

Recall the following theorem :

Theorem: Let $\displaystyle \{f_n\}_n$ be a sequence of $\displaystyle C^1$ functions defined over a connected open set $\displaystyle \mathcal{U}$ of a Banach space E, with values over the Banach space F. We assume that the sequence of differentials $\displaystyle \{f'_n\}_n$ converges uniformly to a function g and that there exists $\displaystyle a \in \mathcal{U}$ such that $\displaystyle \{f_n(a)\}_n$ has a limit in F.

Therefore, the sequence $\displaystyle \{f_n\}_n$ converges uniformly in any ball of $\displaystyle \mathcal{U}$ to a $\displaystyle C^1$ function, and $\displaystyle f'=g$

Q.2

Let $\displaystyle D~:~ \mathcal{C}_0^1(J) \to \mathcal{C}^0(J)$ be the linear mapping $\displaystyle f \mapsto f'$. Prove that :

$\displaystyle \textcircled{a} ~:~ \|D(f)\|_0 \leqslant \|f\|_1$ for any $\displaystyle f \in \mathcal{C}_0^1(J)$

$\displaystyle \textcircled{b} ~:~$ D is bijective

$\displaystyle \textcircled{c} ~:~ |f(t)| \leqslant |t| \cdot \|D(f)\|_0$ for any $\displaystyle f \in \mathcal{C}_0^1(J)$ and for any $\displaystyle t \in J$

$\displaystyle \textcircled{d} ~:~$ D is an isomorphism of Banach spaces from $\displaystyle \mathcal{C}_0^1(J)$ over $\displaystyle \mathcal{C}^0(J)$ (that is to say : the linear mapping $\displaystyle D^{-1}$ is continuous)

Q.3

We define the function $\displaystyle \Phi ~:~ \mathcal{C}_0^1(J) \to \mathcal{C}^0(J)$, $\displaystyle f \in \mathcal{C}_0^1(J) \mapsto \Phi(f)=D(f)-f^2$.

Prove that $\displaystyle \Phi$ is a $\displaystyle C^1$ mapping, calculate its differential and prove that $\displaystyle \Phi'(0)$ (differential of $\displaystyle \Phi$ at the point 0) is an isomorphism.

Q.4

For $\displaystyle r>0$, let $\displaystyle B(0;r)$ be the open ball, center 0, radius r, of $\displaystyle \mathcal{C}^0(J)$.

Prove that there exists $\displaystyle r>0$ and a $\displaystyle C^1$ function $\displaystyle \Psi ~:~ B(0;r) \to \mathcal{C}_0^1(J)$ such that for any $\displaystyle g \in B(0;r)$, $\displaystyle \Psi(g)$ is a solution of the differential equation $\displaystyle y'=y^2+g$

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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)