Let $X,Y$ be Banach spaces, define by $F:X\times Y \rightarrow \mathbb{R}$ be a functional, $F_u,F_v$ be Frechet derivative of $F$ with respect to $u$ and $v$ variables. We show that $F$ is Frechet differentiable.
We have
\begin{align} P&=\Vert F\left(u+h,v+l\right)-F\left(u,v\right)-F_u\left(u,v\right)h-F_v\left(u,v\right)l \Vert \\ &= \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h+F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert +\Vert F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert +\epsilon\Vert l\Vert \end{align} We need to show that $$\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert \leq \epsilon' \Vert h\Vert \quad ???$$
It follows that $F$ admits $F'\left(u,v\right)=F_u\left(u,v\right)h+F_v\left(u,v\right)l$ is the Frechet derivative. Besides, we have that
> $$\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v+l\right)h \Vert \leq \epsilon'' \Vert h\Vert \quad $$ Therefore, \begin{align} Q&=\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert \\ &=\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v+l\right)h +F_u\left(u,v+l\right)h -F_u\left(u,v\right)h \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v+l\right)h \Vert+\Vert F_u\left(u,v+l\right)h -F_u\left(u,v\right)h \Vert \\ &\leq \epsilon''\Vert h\Vert +\Vert F_u\left(u,v+l\right)-F_u\left(u,v\right)\Vert \Vert h \Vert \end{align} On the other hand, since $F_u$ is continuous with respect to $v$ variable it follows that for every $\epsilon''' >0$ there exists a $\delta >0$ such that $\Vert F_u\left(u,v+l\right)-F_u\left(u,v\right)\Vert <\epsilon'''$ for all $\Vert \left(h,l \right) \Vert < \delta.$
Therefore, $$Q \leq \left(\epsilon'' +\epsilon'''\right) \Vert h\Vert.$$
By Cauchy-Schwarz's inequality It leads to that \begin{align} P&\leq \epsilon'\Vert h\Vert+\epsilon \Vert l\Vert , \quad \epsilon'=\left(\epsilon'' +\epsilon'''\right) \\ &\leq \sqrt{{\epsilon'}^2+{\epsilon}^2}\sqrt{\Vert h\Vert^2 + \Vert l \Vert^2}\\ &= \sqrt{2}\epsilon\Vert \left(h,l\right)\Vert, \quad \epsilon=\epsilon' \end{align}
Consequently, $F$ is Frechet differentiable!
Can you tell me whether or not there is any mistake in this proof?
There are two mistakes in this proof, that can be resolved by imposing the assumption of continuity on both $F_u$ and $F_v$.
First in $$\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v+l\right)h \Vert \leq \epsilon'' \Vert h\Vert \quad$$
Here $\delta$ depends on $l$. we know $u,v$ are fix but not $l$ ! as $l$ gets small $\delta $ may get small too.
Second, Absolutely $F_u$ is not linear (necessarily) nor continuous!