Comment by constantcrying
7 hours ago
Most relevant applications use L^2 spaces which can not be defined point wise.
If you want to talk about applications, then this representation is especially bad. Since the intuition it gives is just straight up false.
Fwiw, my favourite textbook in communication theory (Lapidoth, A Foundation in Digital Communication) explicitly calls out this issue of working with equivalence classes of signals and chooses to derive most theorems using the tools available when working in ℒ_2 (square-integrable functions) and ℒ_1 space