Some classes of probabilistic inner product spaces and related inequalities

We give a new definition for probabilistic inner product spaces, which is sufficiently general to encompass the most important class of proba-bilistic inner product spaces (briefly, PIP spaces). We have established certain classes of PIP spaces and especially, illustrated that how to construct a real inner product from a Menger PIP space. Finally, we have established the analogous of Cauchy{Schwarz inequality in this general PIP spaces.