Notably, one very important part which does not go through is that for mappings between infinite dimensional spaces linearity does not imply continuity. (E.g. a

Notably, one very important part which does not go through is that for mappings between infinite dimensional spaces linearity does not imply continuity. (E.g. a series of functions bounded by a constant can have arbitrarily large derivatives) A large part of functional analysis is dealing with that fact and its implication for PDEs.