Why and whence the Hilbert space in quantum theory?

We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, scalar product, orthogonality, linear functional are derivable from statistical description micro-events Hilbertian sum squares $|\mathfrak{a}_1|^2+|\mathfrak{a}_2|^2+\cdots$. latter leads (non-axiomatically) to standard writing Born formula $\mathtt{f} = |\langle\psi|\varphi\rangle|^2$. As a corollary, status Pythagorean theorem, concept length, 6-th problem undergo `revision'. An issue deriving norm topology may no have short-length solution (too many abstract math-axioms) but is likely solvable affirmative; reformulated as mathematical one.