# Math Help - An exercise from my topology & differential calculus exam

1. ## 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
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 $y'=y^2+g$

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

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

3. 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)