Domination of quadratic forms

We study domination of quadratic forms in the abstract setting of ordered Hilbert spaces. Our main result gives a characterization in terms of the associated forms. This generalizes and unifies various earlier works. Along the way we present several examples.
Comment: The article arXiv:1608.06798v1 has been split, the present article contains the first part. The material presented here is partly based on the third named author's master's thesis (arXiv:1604.05114)