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 is bijective. Show that f(U) is an open set in F
then suppose that
show that 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 there exist open sets and such that is a diffeomorphism, in particular every point has a nieghbourhood which means is open.
For the second, check that locally by the argument above.