The Banach Gelfand Triple (S0, L2, S0*)(Rd) (which arose in the
context of Time-Frequency Analysis) is a simple and useful tool,
both for the derivation of mathematically valid theorems AND for
teaching relevant concepts to engineers and physicists (and of
course mathematicians, interested in applications!).
In this context the basic terms of an introductory course on Linear
System’s Theory can be explained properly: Translation invariant
systems viewed as linear operators, which can be described as
convolution operator by some impulse response, whose Fourier
transform is well defined (and is called transfer function), and
there is a kernel theorem: Operators T : S0(Rd) to S0*(Rd)
have a “matrix representation” using some sigma in S0*(R2d) and so on...

This is a modified version of the IWOTA talk 2019 ...