 Let "T" be a distribution on "U", then one can find for every multi-index "α" a continuous function "g""α" such that

and that any compact subset "K" of "U" intersects the supports of only finitely many "g""α"; therefore, to evaluate the value of "T" for a given smooth function "f" compactly supported in "U", we only need finitely many "g""α"; hence the infinite sum above is well-defined as a distribution.If the distribution "T" is of finite order, then one can choose "g""α" in such a way that only finitely many of them are nonzero.The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions.