The tools developed in the previous chapters allow us to define and study in the operator-valued case the various families of functions appearing in classical Schur analysis. In this section we obtain realization formulas for these functions. These formulas in turn have important consequences, such as the existence of slice hyperholomorphic extensions and results in function theory such as Bohr’s inequality. Recall that all two-sided quaternionic vector spaces are assumed to satisfy condition (5.4). An important tool in this chapter is Shmulyan’s theorem on densely defined contractive relations between Pontryagin spaces with the same index, see Theorem 5.7.8, and this forces us to take two-sided quaternionic Pontryagin spaces with the same index for coefficient spaces, and not Krein spaces. The rational case, studied in the following chapter, corresponds to the setting where both the coefficient spaces and the reproducing kernel Pontryagin spaces associated to the various functions are finite-dimensional.