> It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.
Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.
In physics it is common to work explicitily with the components in a base (see tensors in relativity or representation theory), but it's also very important to understand how your quantities transform between different basis. It's a trade-off.
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
Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.