In this paper we provide a study of quaternionic inner product spaces. This includes ortho-complemented subspaces, fundamental decompositions as well as a number of results of topological nature. Our main purpose is to show that a closed uniformly positive subspace in a quaternionic Krein space is ortho-complemented, and this leads to our choice of the results presented in the paper.
D. Alpay, F. Colombo and I. Sabadini. Inner product spaces and Krein spaces in the quaternionic setting., Operator Theory: Advances and Applications, vol. 244 (2015), pp. 33-65.