Abstract:
The kernel theorem, proved by L. Schwarz in 1950, is a fundamental result in
the theory of distributions. It plays a major role in the theory of partial
differential operators. According to J. Dieudonne, it is «probably, the most
important theorem of modern linear functional analysis». Informally speaking,
the theorem says that every continuous linear operator between functional
spaces that satisfy some mild conditions, is an integral operator in a certain
sense (namely, its kernel function is a distribution rather than a genuine
function).
The goal of the talk is to discuss one of the numerous proofs of the kernel
theorem. This proof is due to A. Grothendieck and is based on his theory of
nuclear locally convex spaces (primarily inspired by the Schwarz theorem). All
necessary definitions and preliminaries will be given during the talk.
