If the derivative of a continuously differentiable functions between Banach spaces at a point is an isomorphism, is it so also in a neighborhood?

Question

Let $X$ be a Banach space, $U$ an open subset of $X$ and $f:U\to X$ a Fréchet continuously differentiable function. Suppose that, for a point $p\in U$, $Df(p):X\to X$ is an isomorphism. Does there exist a neighborhood $V$ of $p$ such that $Df(x)$ is an isomorphism for all $x$ in $V$?

I tried to prove this by contradiction, but I ended up extracting a convergent subsequence from a bounded sequence, a thing that you can't do in general Banach spaces.

Answer 1

Yes. This follows immediately from the fact that if $T$ is invertible and $\|S-T\|<\frac 1 {\|T^{-1}||}$ then $S$ is invertible.