I encountered some gaps in Amann and Escher’s Analysis I
the hypothesis of theorem3.2 is the function f is n-times continuously differentiable and its domain is a convex perfect subset of $\mathbb{K}(\mathbb{R} or \mathbb{C})$ and its codomain is a Banach space，the first place is about how to use the estimate about the remainder function to derive the corollary3.3 which says $\frac{∥R_{n}(f,a)(x)∥}{|x-a|^n}→0(x→a)$，in other words we need to show that $\text{sup}_{t\in(0,1)}(f^{(n)}(a+t(x-a))-f^{(n)}(a))→0(x→a)$
we know that the two variable(x and t)function inside is continuous(because the continuity of norm and n th derivative),but is this function after taking supremum over $t\in (0,1)$still continous(about x)? I try to prove this but fail
the second place is a similiar problem,after using the mean value theorem to get the inequality,and then taking limit $n→\infty$,how can we deduce (2.1)?I think again the author interchange the order of supremum and limit just like corollary3.3,and after getting (2.1) he does the same thing again but he doesn't prove this.