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Thread: banach space

  1. #1
    mms
    mms is offline
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    banach space

    Let E,F,G be 3 banach spaces, U an open set of E and f:U->F , g:U->G be 2 continously differentiable applications.

    Suppose that $\displaystyle
    \forall x \in U\,Df(x)$ is bijective. Show that f(U) is an open set in F

    then suppose that $\displaystyle
    \exists \varphi :f(U) \to G\,such\,that\,g = \varphi \circ f
    $

    show that $\displaystyle \varphi $ is continously differentiable
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  2. #2
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    Quote Originally Posted by mms View Post
    Let E,F,G be 3 banach spaces, U an open set of E and f:U->F , g:U->G be 2 continously differentiable applications.

    Suppose that $\displaystyle
    \forall x \in U\,Df(x)$ is bijective. Show that f(U) is an open set in F

    then suppose that $\displaystyle
    \exists \varphi :f(U) \to G\,such\,that\,g = \varphi \circ f
    $

    show that $\displaystyle \varphi $ is continously differentiable
    For the first one apply the inverse function theorem (This applies because a bounded linear transformation has a bounded inverse iff it's bijective by a simple application of the closed graph theorem) to get for every $\displaystyle x \in U$ there exist open sets $\displaystyle x\in U_{1,x} \subset U$ and $\displaystyle U_{2,x} \subset f(U)$ such that $\displaystyle f: U_{1,x} \rightarrow U_{2,x}$ is a diffeomorphism, in particular every point $\displaystyle f(x)$ has a nieghbourhood $\displaystyle U_{2,x} \subset F(U)$ which means $\displaystyle f(U)$ is open.

    For the second, check that $\displaystyle \varphi \in C^1$ locally by the argument above.
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