Let $U$ and $V$ be open subsets of a Banach space $E$ such that $0\in U\cap V$ If a diffeomorphism $f\colon U\to V$ has $0$ as a hyperbolic fixed point, then $f$ and $Df(0)$ are locally topologically conjugate at $0$ i.e. there are neighborhoods $\tilde U$ and $\tilde V$ of $0$ and a homeomorphism $h\colon \tilde U\to \tilde V$ such that $Df(0)h = hf$