Suppose $F:X \to Y$ is a map from Banach spaces $X=\widetilde{C}^{k+2, \alpha}(S)$ to $Y = \widetilde{C}^{k, \alpha}(S)$ where $S = I \times [0,T].$
Suppose the derivative $DF(u):X \to Y$ exists and is continuous.
(1) Am I right that the statement
$DF[u]^{-1}$ are uniformly bounded for bounded $u$
means $\lVert DF[u]^{-1}\rVert \leq M$ holds for all bounded (in what?) $u$ and $M$ is a constant not depending on $u$?
(2) If I write $DF[u]h = f$, and get a bound $\lVert h \rVert \leq C\lVert h_0\rVert + C\lVert f \rVert$ where $h_0 = h(\cdot, 0)$, then how does this show that $DF[u]^{-1}$ is uniformly bounded? (If the constant doesn't depend on $u$).
(3) Willie Wong said that if the $DF[u]^{-1}$ are uniformly bounded then applying the inverse function theorem to F, the size of the neighbourhood of $F(u^0)$ that is invertible doesn't depend on $F(u^0)$. Can someone give me a reference for this fact?
Thanks.
(These questions stem from Inverse function theorem in Banach space to prove short time existence of PDE (explanation of statements), and I didn't want to keep asking questions there!)