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 ?)