Define the functional $$F[y] = \int_0^1 f(x,y)p(x) \, dx$$ 1)What is the domain and range of the functional if $p(x) = 1$? 2)How does it change for more general functions $ρ$? My work:
1) The domain is all $C^0$ functions defined on $[0,1]$ and range is all real numbers since the integral is with respect to $x$
2) I'm guessing that the domain and range only changes from 1) if the there exists discontinuity in the integral of $p(x)$
To get things in order... I guess you mean that $y\in [0,1]$ is fixed and $F_y$ is the functional mapping $f$ to the integral $\int_0^1 f(x,y)p(x)\,dx$. Is this right? Well, in this case, $f$ is surely not a function on $[0,1]$, but on $[0,1]^2$. However, since $y$ is fixed, we can identify $f$ with a function on $[0,1]$. If $p=1$ - with this identification in mind - the (maximal) domain is $L^1(0,1)$. Without identification, the domain is $L^1((0,1)\times\{y\})$. And of course, the range is $\mathbb R$.