#hyperfunction

[isomorphismes]

Test functions and [tempered] distributions require the notion of topological vector space … distributions can be traced back to Green's functions in the 1830’s to solve ordinary differential equations … the 1936 work of Sergei Sobolev on hyperbolic PDE’s.

Laurent Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on.

tempered distribution

topological vector space

(^ if there is a dipole, there must be a notion of subtraction, hence the need for a vector, and to speak of this very conceptually, use a TVS)