Importance of Schwartz kernel theorem I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that would not have been possible without it?
Why are we so interested in trying to represent operators as 'integral operators'? An answer which emphasizes useful interpretation of the theorem in the context of physics or engineering is most welcome.
Thanks.
(Also, are there theorems which tell us exactly when the kernel is a locally integrable function ?)
 A: The main reason for the importance of the Schwartz Kernel Theorem, in my opinion, is that it allows one to do multilinear and tensor algebra on infinite dimensional spaces like $\mathcal{S}(\mathbb{R}^d)$, $\mathcal{S}'(\mathbb{R}^d)$ with the same ease and flexibility as for finite dimensional spaces.
Also, an example of application to probability is as follows.
Suppose you have a Borel probability measure $\mu$ on $\mathcal{S}'(\mathbb{R}^d)$.
Suppose that for every test functions $f_1,\ldots,f_n$ in $\mathcal{S}(\mathbb{R}^d)$,
the function $\phi\mapsto \phi(f_1)\cdots \phi(f_n)$ is in $L^1(\mathcal{S}'(\mathbb{R}^d),\mu)$. Then you get a continuous multilinear map
$$
S_n:\mathcal{S}(\mathbb{R}^d)^n\longrightarrow \mathbb{R},
$$
$$
(f_1,\ldots,f_n)\mapsto \int_{\mathcal{S}'(\mathbb{R}^d)}\phi(f_1)\cdots \phi(f_n)
\ d\mu(\phi)
$$
The nuclear theorem then associates to this map a unique distribution $T\in \mathcal{S}'(\mathbb{R}^{nd})$ such that
$$
T(f_1\otimes\cdots \otimes f_n)=S_n(f_1,\ldots,f_n)
$$
for all test functions. You can then ask properties of $T$, like what is the singular support, etc. which inform you about the moment $S_n$ of the measure
and ultimately will tell you something about this measure itself.
A: The theorem is very important because it allows to represent a continuous operator through just a distribution (that is its kernel), which is much easier.
It is like in the spectral theory: every selfadjoint operator can be represented by its spectral function. Studying a function is much more easier than a operator......
