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

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- June 6th 2010, 09:00 AMmmsbanach 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 is bijective. Show that f(U) is an open set in F

then suppose that

show that is continously differentiable - June 24th 2010, 05:45 PMJose27
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.