Bounded $q$-variation Functions in Spaces with Indefinite Metric

In this paper we establish the definition of bounded q-variation functions on Krein spaces (definition 7) and give a complete characterisation by comparing it with bounded q-variation functions on classical Hilbert spaces and with Hilbert spaces associated to a Krein space (theorem 6). The fundamental tools of the theory of bounded q-variation functions in the formalism of Krein spaces are described (theorems 7, 8). Furthermore, we endow the spaces of bounded q-variation functions in Krein spaces with a norm, which we have called q-norm (theorem 13).