I doubt whether any other process can make a more effective use of Hilbert space structure.
Given several vectors, which may be or not be orthogonal to each other, how to construct a vector perpendicular to them? In a sense of inner production being zero.
To be specific, I have vectors of length , where . So the maximum rank for these vectors is , which leaves space for new vectors perpendicular to all of them. How to construct such a vector? I know Gram-Schmidt process, but it seems it's not what I want.