Assume a function $f:L_2(R^{+}):R$ is frechet differnetiable in $x\in L_2(R^{+})$ in that there exists a unique function $D(x_i,x)$ (where $x_i\in R^{++}$ is an element of x) such that:
$f(x+h)=f(x)+\int_i D(x_i,x)h_idi + o (||h||_2)$
Further assume we can define the second cross partial, $DD(x_i,x_j,x)$ in a similar way. Do we then have that $\int_{-\infty}^{x_j} DD(x_i,y,x) dy= D(x_i,x)+C?$
It seems like there would be 2 ways to interpret $\int_{-\infty}^{x_j} DD(x_i,y,x) dy$.
The first being what feels like the "typical" sense, where the value of the second item ($y$ or $x_j$) increases along the integral, but also the $x_j$ that lives inside of $x$ increases too, making it seem analogous to the typical partial derivative leading to my feeling the fundamental theorem of calculus holds.
The second interpretation is to take the integral while ignoring the fact that $x_j$ is an element of $x$, which seems to suggest FTC would not necessarily hold.
Both interpretations seem like they could arise in certain contexts, but I am not aware of an explicit discussion of the second, though it arises in an application I have due to the way a limit is constructed. Is there a name or way of identifying the different ways of taking the integral, and does anyone know of any resources to help with understanding the second?