> [!theorem] > > Let $\cx$ be a [[Banach Space|Banach space]] and $F: \cx \to \cx$ be any mapping. Suppose that $\text{Id} + F$ is a [[Contractor|contraction]], then $F$ is [[Homeomorphism|invertible]]. > > *Proof*. Let $y \in F$, and take $g_y(x) = (\text{Id} + F)(x) - y$, then $F(x) = y$ if and only if $g_y(x) = x$, and $g_y$ is also a contraction. By the [[Contractor|Banach fixed-point theorem]], $g_y$ has a unique fixed point. Therefore there exists $x \in E$ such that $y = F(x)$. > [!theorem] > > Let $\cx$ be a Banach space, $F: \cx \to \cx$ be any mapping. Suppose that there exists a [[Homeomorphism|homeomorphism]] $H: \cx \to \cx$ such that $\text{Id} + HF$ is a contraction, then $F$ is invertible. > > *Proof*. Since $HF$ is invertible and $H$ is a homeomorphism, $F$ is invertible as well.