Application of Schwartz Kernel Theorem to Quantum Mechanics

Application of Schwartz Kernel Theorem to Quantum Mechanics

I am currently reading Quantum Mechanics for Mathematicians and have a
question about a statement made in the book:
Remark. By the Schwartz kernel theorem, the operator B can be represented
by an integral operator with distributional kernel $K(q,q')$. Then the
commutativity $BQ = QB$ implies that, in the distributional sense,
$$ (q-q')K(q,q') = 0, $$
so that $K$ is "proportional" to the Dirac delta-function, i .e., $$
K(q,q') = f(q)\delta(q-q'). $$ This argument is usually given in physics
textbooks.
$B$ and $Q$ are both operators in the coordinate representation. $Q$ is
the position operator and $B$ is bounded.



So, first of all, I can't find much about the kernel theorem online. I've
been using this as a reference. According to this document, there is a
relation between bilinear forms and distributions. However, I don't know
how to view $B$ as a bilinear form and thus apply the theorem. Can someone
elucidate how the kernel theorem is applicable to $B$? Secondly, what
arguments/notation in physics is the author (of the QM book) referencing?