I am making my way through Gelfand's book. I encountered the lemmas that build towards the derivation of Euler-Lagrange. Starting from a most general example of a linear functional:
$$ \phi[h] = \int_{a}^{b}\left[ \alpha_{0}(x)h(x) + \alpha_{1}(x)h'(x) + \alpha_{2}(x)h''(x) + \dots \right]dx $$
with $\alpha_i(x)$ fixed functions in $\mathscr{C}(a,b)$, the author proceeds to determine the form of the $\alpha$'s that would make all $h(x)$ in a class vanish. I.e., what is the form, say, of $\alpha_2(x)$ that makes $\int_{a}^{b}\alpha(x)h''(x)dx = 0$ for all $h(x)$ in $\mathscr{D}_2(a,b)$, etc.
Here is where I encounter this:
$$ h(x) = \int_{a}^{x}d\xi \int_{a}^{\xi}\left[ \alpha(t) -c_0 - c_1t \right]dt $$
It looks to me like $h(x)$ is a twice differentiable function. We don't know its shape except that the second derivative looks like $\alpha(x) -c_0 - c_1x$ for some function $\alpha(x)$ and constants $c_0$ and $c_1$. Am I getting this right?
But then why does the double integration look like this? It's not exactly a double integration, more like twice integrating on the same independent variable, right? In general, in my previous experience, the notation for a double integration of a function $f(x, y)$ was like $\int\int_Rf(x,y)dydx$ or some variation thereof.
So, what's up with this notation?
