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I am trying to get my head around these introduction to Hilbert/Banach spaces. I kind of get the basic, but is still currently struggle to prove the following question.

Suppose $F(x)=0$ is a non-linear forward operator acting on a pair of Hilbert function spaces $F: H_1 \rightarrow H_2$ which has the following properties: $$ \| F'(x)\|_{\mathcal{L}(H_1, H_2)} \le N_1 \forall x \in H_1 $$ $$ \| F'(x) - F'(y)\|_{\mathcal{L}(H_1, H_2)} \le N_2 \|x - y\| \forall x,y \in H_1 $$ where $F'(x)$ is the Fréchet derivative. Using the Taylor's formula: $$ F(x+h) = F(x) + F'(x)h + G(x,h) $$ show that for the remainder: $$ \|G(x, h)\|_{H_2} \leq \frac{1}{2}N_2 \|h\|^2_{H_1} $$

I know it should be something pretty straightforward (integration by parts was on my bet list) but I apparently couldn't converge to an appropriate answer.

Quang Thinh Ha
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Let $u\in H_2$ be a unit vector such that $\langle G(x,h), u\rangle = \|G(x,h)\|$. Consider the function $$f(t) = \langle F(x+th), u\rangle $$ Since $f'(t) = \langle F'(x+th) h, u\rangle$, it follows that $|f'(t)|\le N_1\|h\|$ and $|f'(t)-f'(s)|\le N_2\|h\|^2 |t-s|$. I don't think we need the first of these, actually.

The function $g(t) = f(t) - f(0) - f'(0)t$ satisfies $g(0)=0$ and $|g'(t)|\le N_2\|h\|^2 t$, hence $$ |g(1)| \le \int_0^1 |g'(t)|\,dt \le \int_0^1 N_2\|h\|^2 t\,dt = \frac12 N_2\|h\|^2 $$
which proves the claim because $$ g(1) = f(1)-f(0)-f'(0) = \langle F(x+h)-F(x)-F'(x)h, u \rangle = \|G(x,h)\| $$