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> 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.

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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.

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

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