Differential range invariance for Banach-space maps

Let $X$, $Y$, and $Z$ be real Banach spaces, let $U\subseteq X$ be nonempty, let $x_0\in U$, and let $F:U\to Y$ be a map. A differential range-invariance factorization at $x_0$ on $U$ consists of a bounded linear map $K:Z\to Y$ and a map $r:U\to Z$ such that

$F(x)-F(x_0)=Kr(x)(x\in U)\mathrm{.}$

Thus every data increment lies in the range of the fixed operator $K$. The factorization alone does not assert that $K$ is injective, that $r$ is unique, or that a tangential-cone inequality holds. If $U$ is open and $F$ and $r$ are Fréchet differentiable, differentiating this identity gives $F^{\mathrm{\prime }}(x)=K{r}^{\mathrm{\prime }}(x)$ at every $x\in U$; no converse is asserted without further hypotheses.

Real Banach spaces and their norm metrics

The distinct local Hilbert-space tangential cone condition

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equation (1.6)