I am reading Jaynes' "probability theory: the logic of science" which refers at one point to Dirac's delta function an example of a so called "generalized function".
Jaynes then says that "mathematicians thinking in terms of the set theory definition of a 'function'" considered delta-functions non-rigorous.
He explains how Lighthill justifies delta-functions/generalized functions using a different basis than set theory.
My question is: What is a good resource on (or could you explain) how to define functions in this non-set-theory way of Lighthill, and how the delta function is justified on the basis of it?
I am not very familiar with Fourier Analysis.
Here is the entire section in Jayne's book, in case it helps:
For references try: "Delta Functions: Introduction to Generalized Functions", R.F. Hoskins; "Mathematics for the Physical Sciences", Laurent Schwartz; "Generalized Functions vol 1", I.M. Gelfand, G. E. Shilov. The last 2 are classics in the areas of generalized functions and distributions.
But basically $δ(x)$ or $δ(x-a)$ is defined by
$$f(0)=\int f(x)δ(x)dx$$
or
$$f(a)=\int f(x)δ(x-a)dx ,$$
Where f is adequately well behaved. Either suffices. So δ(.) is usually defined by its action under the integral sign. Sometimes, in addition, its value is given as zero everywhere except at the single point, a, where its value is infinity, but this may not be needed.
However there are problems with this definition since a value at a single point should not change the value of an integral. So instead sometimes a 'inner product' notation is used
$$f(a) = (f(x) , δ(x-a)) $$
or
$$f(a) = <f(x) , δ(x-a)> $$
which transforms $f(x)$ to $f(0)$, or $f(x)$ to $f(a)$ --see Gelfand p4 or Schwartz p77,
Also it is better to use the terms 'Delta Distribution' or 'Dirac Delta' since $δ(x-a)$ is not properly a function.