Show that the following functional is Frechet differentiable

Question

I am new to this and I need to show that the following functional is Frechet differentiable:

\begin{equation} f(u) = \sin(u(1)) \ \ in \ \ C[0,1] \end{equation}

What I have already done:

\begin{equation} \lim_{t \to 0} \frac{\|F(u+h)-F(u)-Ah\|_{V}}{\|h\|_{U}} = 0 \end{equation}

Gateaux derivative: $$\begin{split} A = F'_{G}(u)=\lim_{t \to 0} \frac{F(u+th)-F(u)}{t} = \lim_{t \to 0} \frac{\sin(u(1)+th(1))-\sin(u(1))}{t} \\ = \frac{d}{dt} \sin(u(1)+th(1)) \arrowvert_{t=0} = \cos(u(1)+th(1))\cdot h(1) \arrowvert_{t=0} \\ = \cos(u(1))\cdot h(1) \end{split} $$

Substituting $A$ back: $$ \lim_{t \to 0} \frac{\|F(u+h)-F(u)-\cos(u(1))\cdot h(1)\|_{V}}{\|h\|_{U}} = \lim_{t \to 0} \frac{\|\sin(u(1)+h(1))-\sin(u(1))-\cos(u(1))\cdot h(1)\|_{V}}{\|h\|_{U}} $$ That's it, I don't know how to continue... I would appreciate any help.

Answer 1

For every $u, h \in C([0,1])$ we have $$ f(u+h)=\sin(u(1)+h(1))=\sin(u(1))+h(1)(\sin)'(u(1))+\frac12h^2(1)(\sin)''(u(1))+o(h^2(1)), $$ i.e. $$ f(u+h)=f(u)+h(1)\cos(u(1))-\frac{h^2(1)}{2}\sin(u(1))+o(h^2(1)). $$ Since $$ |h(1)| \le \|h\|_\infty \to 0\, \text{ as }\, h \to 0, $$ it follows that $$ f(u+h)=f(u)+h(1)\cos(u(1))+o(\|h\|_\infty). $$ Thus $f$ is Fréchet differentiable and $$ Df(u)\cdot h=h(1)\cos(u(1)) \quad \forall u, h \in C([0,1]). $$