Why and whence the Hilbert space in quantum theory?

Abstract

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