Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ $U\subset B$ and u+h lies entirely within U. If F is $C^{k}$ then
$F(u+h)=F(u)+dF(u;h)+\frac{1}{2!}d^2F(u;h)+\dots+\frac{1}{(k-1)!}d^{k-1}F(u;h)+R_k$ ,where the remainder term is given by $R_k(u;h)=\frac{1}{(k-1)!}\int_0^1(1-t)^{k-1}d^kF(u+th;h)\,dt$ and $d^{k}F(u+t, h)$ is the kth Gateaux derivative of $F(u+th)$ in the $h$ direction.
Question: We want to show $lim_{h\to u}R_{k}(u,h)=0$. Any references on the proof (ideally books and links I can access online)? Any suggestions on how to prove it?
The complex Gateaux derivatives are for $\psi=\psi_{1}+i\psi_{2}$
$dF(\psi,\xi)=d_{\psi_{1}}F(\psi_{1},\xi_{1}-id_{\psi_{2}}F(\psi_{2},\xi_{2}$ and $\bar{d}F(\psi,\xi)=d_{\psi_{1}}F(\psi_{1},\xi_{1}+id_{\psi_{2}}F(\psi_{2},\xi_{2}$. So it follows immediately because of linearity of integrals.
Thank you
You have the assumption that $F$ is $C^k$. Hence, for every $\delta > 0$, there is $\varepsilon > 0$, such that $$\| d^k F(\tilde u; h) - d^k F(u; h)\| \le \delta$$ for all $\tilde u$ in an $\varepsilon$-ball around $u$.
Can you continue?