X is the vector space $X = C([1-,1],\mathbb{R})$ equipped with sup-norm. For $j = 1,\cdot\cdot\cdot,n$ let $a_j\in \mathbb{R}$ and let $x_j \in [-1,1]$. Let $F:X\to \mathbb{R}$ be defined by
$F(f) = \sum\limits_{j=1}^na_j\cdot f(x_j)^j$.
I'm to show that F is differentiable at each element $f\in X$ and find the derivatives. But I'm having a bit of trouble getting started.
I think the way to go is to "guess" at a derivative, ie just use the directional derivate of $F(f(y))$ and then check that this candidate actually is a derivative, ie check/prove that it is linear, continuous, bounded and that the rate of convergence equals zero (ie $\underset{r\to 0}{lim} \frac{||\sigma(r)||}{||r||} = 0$). And then, if I've understood things correctly, I'd be done.
So... Am I supposed to being solving this problem with something like:
We take an arbitrary $f\in X$. Then we pick a point, b$\in [-1,1]$ and try to calculate the directional derivative of F in the direction of some r:
$F(f(b,r)) = \underset{t\to 0}{lim} \frac{F(f(b+tr)-F(f(b))}{t}= \underset{t\to 0}{lim} \frac{a_j\cdot f(x_j+tr)^j-a_jf(x_j)^j}{t} $. (can write $x_j$ instead of b bc every b will correspond to one $x_j$ value and bc when t goes to zero, we don't have to worry about switching which part of the partition we're in).
I'm assuming I've done it wrong here, but regardless I'm lost as to how to proceed or start properly solving it.
Obviously all of what I've written is just spiderwebs and hogwash, but I'd be grateful if anybody could nudge me in the right direction or provide a hint of some sort.
$$ F(f + h) - F(f) = \sum\limits_{j=1}^na_j\cdot (f + h)(x_j)^j - \sum\limits_{j=1}^na_j\cdot f(x_j)^j = $$ $$ = a_1(f + h)(x_1) - a_1(f)(x_1) + 2a_2 f(x_2)h(x_2) + 3 a_3 f(x_3)^2 h(x_3) + \cdots + n a_n f(x_n)^{n-1} h(x_n) +\hbox{higher order terms} = $$ $$ = a_1 h(x_1) + 2a_2 f(x_2)h(x_2) + 3 a_3 f(x_3)^2 h(x_3) + \cdots + n a_n f(x_n)^{n-1}h(x_n) + \hbox{higher order terms} $$ I.e., $DF(f)$ is the linear function $X\longrightarrow\Bbb R$ $$h\longmapsto a_1 h(x_1) + 2a_2 f(x_2)h(x_2) + 3 a_3 f(x_3)^2 h(x_3) + \cdots + n a_n f(x_n)^{n-1}h(x_n)$$