mathematical physics - Quantum mechanics in a metric space rather than in a vector space, possible?

Quantum mechanics starts with wave functions living in Hilbert space. But later for Born's interpretation, the wave function need to be of unit energy (I mean total probability = 1, $\int_{-\infty}^{\infty}|\psi(x)|^2dx = 1$). But for two elements of Hilbert space when summed to get a third element in the same space, the third need not be of unit energy. So these two assumptions mathematically inconsistent. Either you have to throw away Hilbert space or the Born's interpretation.

If we throw away Hilbert space, assume $\psi$ to be of unit energy and is an element of a metric space (rather than a vector space) with suitable metric defined on it. Would this kind of an idea which makes more mathematical sense would lead to a physical theory. Right now the mathematics of QM is simply ad-hoc.