I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions.
Let $L_2(a,b)$ be the Hilbert space of the square-integrable functions on $(a,b)$ and let $A$ be a limited linear operator of the form
$(Af)(x)=\int_a^bK(x,t)f(t)dt$.
My professor said that in these hypotesis the integral kernel $K(x,t)$ is a generalized function and gave me the example of the identity operator:
$(If)(x)=f(x)=\int_a^bdt \delta(x-t) f(t)$.
I am a bit confused because $\delta$ isn't a continuous linear functional on $L_2(a,b)$. Can I use it as an integral kernel? And if I can in what sense is it a distribution? On which Frechet space? Can I define $\delta$ on a Frechet scpace on which it isn't continuous?
This is of course some kind of abuse of notation that happens often in physics.
One way to make sense of it is to replace $δ$ by an approximating sequence of continuous functions $δ_n$ with the usual properties of pointwise convergence to zero outside the origin and integral $1$ and read
$$I(f)(x)=\lim_{n\to\infty}\int_a^b δ_n(x-t)\,f(t)\,dt.$$