- In the following, $\mathbb K$ denotes $\mathbb C$ or $\mathbb R$ and $E$ is a $\mathbb K$-Banach space
I have known that for a function $f:\mathbb K\supseteq X\to E$ is called differentiable at $a$ if there is a $T_a\in E$ such that $$ \lim_{x\to a}\frac{f(x)-f(a)-T_a\cdot(x-a)}{x-a}=0 $$
A function $f:\mathbb K\supseteq X\to E$ is differentiable at $a$ if there is an $T_a\in\mathcal L(\mathbb K,E)$ such that $$ \lim_{x\to a}\frac{f(x)-f(a)-T_a(x-a)}{|x-a|}=0 $$
It's obvious that for a function $f\in E^{\mathbb K}$, $f$ is differentiable at $a$ iff it is Fréchet-differentiable at $a$
My question is whether $f\in C^n(X,E)$ for "ordinary" differential iff $f\in C^n(X,E)$ for Fréchet-differential.
I tried to prove it by induction but failed. This is my process:
- Define an isometric isomorphism by $$ A_0^{-1}:\mathcal L(\mathbb K,E)\to E,\quad L\mapsto L1 $$ then $A_0$ is $(A_0^{-1})^{-1}$
- Define $A_1:\mathcal L(\mathbb K,E)\to \mathcal L(\mathbb K,\mathcal L(\mathbb K,E))$ by $f\mapsto A_0\circ f$, and iteratively define isomorphisms by $$ A_m:\mathcal L(\mathbb K,\mathcal L^{m-1}(\mathbb K,E))\to L(\mathbb K,\mathcal L^{m}(\mathbb K,E)),\quad f\mapsto A_{m-1}\circ f $$
- I tried to prove (using $f^{(n)}$ for ordinary derivative for distinction) if $$ \lim_{x\to a}\frac{f^{(n)}(x)-f^{(n)}(a)-T_a\cdot(x-a)}{x-a}=0 $$ (using $\partial^nf$ for Fréchet-derivative for distinction) then $$ \lim_{x\to a}\frac{\partial^nf(x)-\partial^nf(a)-(A_n\circ\cdots\circ A_1\circ A_0T_a)(x-a)}{|x-a|}=0 $$ and vice versa, which means $\partial^{n+1}f(a)=A_n\circ\cdots\circ A_1\circ A_0f^{(n+1)}(a)$
but I cannot go further.