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

  1. #1
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    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

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Haha, good luck
    (sidenote : we had a total of 3 hours, and this exercise is worth 1/3 of the marks)
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  2. #2
    Member Last_Singularity's Avatar
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    Just curious, what are the prerequisites for solving these types of questions? Linear algebra, analysis, what else?
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  3. #3
    Moo
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    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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