# Thread: banach space

1. ## 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

2. Originally Posted by mms
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.