# Diffeomorphisms between a nonopen set and an open set of Banach spaces

Let $$E_i$$ be a $$\mathbb R$$-Banach space, $$U_2\subseteq E_2$$ be open and $$f$$ be a $$C^1$$-diffeomorphism from $$B_1$$ onto $$U_2$$.

What do we need to assume in order to conclude that $$B_1$$ is open?

By definition, $$f=\left.\tilde f\right|_{B_1}\tag1$$ for some $$\tilde f\in C^1(U_1,E_2)$$ for some $$E_1$$-open neighborhood $$U_1$$ of $$B_1$$ and $$g:=f^{-1}\in C^1(U_2,E_1)\tag2.$$ Now, $$g(U_2)=B_1\subseteq U_1$$ so that the chain rule is applicable and yields $$\operatorname{id}_{E_2}={\rm D\left(\tilde f\circ g\right)(x_2)={\rm D}\tilde f(g(x_2))\{\rm D}g(x_2)\;\;\;\text{for all }x_2\in U_2\tag3.$$ Now let $$x_1\in B_1$$ and $$x_2:=f(x_1)$$. By $$(3)$$, $${\rm D}g(x_2)$$ has a left-inverse. Assuming $$E_1$$ is finite-dimensional, this implies that $$\operatorname{rank}{\rm D}g(x_2)\ge d:=\dim E_1\tag4;$$ hence $$\operatorname{rank}{\rm D}g(x_2)=d$$.

Now, in order to apply the inverse function theorem, we need that $${\rm D}g(x_2)$$ is bijective. This will clearly hold if $$\dim E_2=d$$. In that case the inverse function theorem yields that $$\left.g\right|_{\Omega_2}$$ is a $$C^1$$-diffeomorphism from an $$E_2$$-open neighborhood $$\Omega_2\subseteq U_2$$ of $$x_2$$ onto an $$E_1$$-open neighborhood $$\Omega_1$$ of $$g(x_2)=x_1$$. And since $$\Omega_1=g(\Omega_2)\subseteq g(U_2)=B_1$$, this shows that $$B_1$$ is open.