An exercise from my topology & differential calculus exam

**Moo** · December 12th 2008, 11:37 PM

Hi !
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 $J=[-1,+1]$ .
Let's denote $\mathcal{C}^0(J)$ the Banach space of continuous functions over J with values in $\mathbb{R}$ with the norm : $\|f\|_0=\sup_{t \in J} |f(t)|$ (uniform convergence norm).
We'll say that if $f \in \mathcal{C}^0(J)$ is $C^1$ (that is continuous and differentiable) if f is $C^1$ over $]-1,+1[$ and if its derivative $f'$ has a limit in -1 and 1. We'll also denote $f'$ the function we then get.
Let's denote $\mathcal{C}^1(J)$ the vector subspace of $\mathcal{C}^0(J)$ made of the $C^1$ functions over J. We associate to $\mathcal{C}^1(J)$ the following norm : $\|f\|_1=\|f\|_0+\|f'\|_0$ .
Let's denote $\mathcal{C}_0^1(J)$ the vector subspace of $\mathcal{C}^1(J)$ made of the functions f such that $f(0)=0$ , with the relative norm.

Q.1
Prove that $\mathcal{C}^1(J)$ and $\mathcal{C}_0^1(J)$ are Banach spaces.
Recall the following theorem :
Theorem : Let $\{f_n\}_n$ be a sequence of $C^1$ functions defined over a connected open set $\mathcal{U}$ of a Banach space E, with values over the Banach space F. We assume that the sequence of differentials $\{f'_n\}_n$ converges uniformly to a function g and that there exists $a \in \mathcal{U}$ such that $\{f_n(a)\}_n$ has a limit in F.
Therefore, the sequence $\{f_n\}_n$ converges uniformly in any ball of $\mathcal{U}$ to a $C^1$ function, and $f'=g$

Q.2
Let $D~:~ \mathcal{C}_0^1(J) \to \mathcal{C}^0(J)$ be the linear mapping $f \mapsto f'$ . Prove that :

$\textcircled{a} ~:~ \|D(f)\|_0 \leqslant \|f\|_1$ for any $f \in \mathcal{C}_0^1(J)$
$\textcircled{b} ~:~$ D is bijective
$\textcircled{c} ~:~ |f(t)| \leqslant |t| \cdot \|D(f)\|_0$ for any $f \in \mathcal{C}_0^1(J)$ and for any $t \in J$
$\textcircled{d} ~:~$ D is an isomorphism of Banach spaces from $\mathcal{C}_0^1(J)$ over $\mathcal{C}^0(J)$ (that is to say : the linear mapping $D^{-1}$ is continuous)

Q.3
We define the function $\Phi ~:~ \mathcal{C}_0^1(J) \to \mathcal{C}^0(J)$ , $f \in \mathcal{C}_0^1(J) \mapsto \Phi(f)=D(f)-f^2$ .
Prove that $\Phi$ is a $C^1$ mapping, calculate its differential and prove that $\Phi'(0)$ (differential of $\Phi$ at the point 0) is an isomorphism.

Q.4
For $r>0$ , let $B(0;r)$ be the open ball, center 0, radius r, of $\mathcal{C}^0(J)$ .
Prove that there exists $r>0$ and a $C^1$ function $\Psi ~:~ B(0;r) \to \mathcal{C}_0^1(J)$ such that for any $g \in B(0;r)$ , $\Psi(g)$ is a solution of the differential equation $y'=y^2+g$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Haha, good luck

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

**Last_Singularity** · January 3rd 2009, 08:18 AM

Just curious, what are the prerequisites for solving these types of questions? Linear algebra, analysis, what else?

**Moo** · January 3rd 2009, 10:26 AM

Hmm well, topology (which uses analysis and linear algebra, but it's a topic in itself) and differential calculus (the basics, like knowing what a differential is)

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