This is a follow-up of the question I posed here https://math.stackexchange.com/questions/1084883/chomsky-feynman-thom
I was reading the following comment on Von Neumann (see below) and it occurred to me that what I am asking about Chomsky and Feynman might be, from a methodological standpoint, an instantiation of the same logic of generalization mentioned there:
[...] This urged Von Neumann to limit the scope of acceptable sequences or functions in both theories to what we know today as spaces ℓ2 and L2 (Von Neumann named them FZ and FΩ, respectively). The Riesz-Fischer theorem, well known to mathematicians since 1907, proved that both spaces were isomorphic and isometric. Given that Von Neumann reasoned that FZ and FΩ (and not Z and Ω!) form the "real analytic substrate" of the matrix and wave mechanics, respectively, and both spaces are isomorphic, the isomorphism means that both theories should yield the same results.
Do you think that Riesz-Fischer theorem (I do not know the specific details of it) might have important consequences for the study of languages and / or for the translation of its results from one kind of formulation to another?
Thanks in advance.
To answer your question positively would require that there is a theory of linguistics which uses functional analysis to model that theory. Otherwise one can not apply that theorem.