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Math Help - banach space

  1. #1
    mms
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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 <br />
\forall x \in U\,Df(x) is bijective. Show that f(U) is an open set in F

    then suppose that <br />
\exists \varphi :f(U) \to G\,such\,that\,g = \varphi \circ f<br />

    show that \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 <br />
\forall x \in U\,Df(x) is bijective. Show that f(U) is an open set in F

    then suppose that <br />
\exists \varphi :f(U) \to G\,such\,that\,g = \varphi \circ f<br />

    show that \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 x \in U there exist open sets x\in U_{1,x} \subset U and U_{2,x} \subset f(U) such that f: U_{1,x} \rightarrow U_{2,x} is a diffeomorphism, in particular every point f(x) has a nieghbourhood U_{2,x} \subset F(U) which means f(U) is open.

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