Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics – have a common underlying mathematical structure. That common structure is Hilbert space.

These two theories are said to be isomorphic.

If you assume the space you’re using has a complex inner product, and that it is a complete metric space under that inner product, is what makes it a Hilbert space. SE Wave Mechanics and Matrix Mechanics have this complex inner product and completeness.