We discuss the notions of strong convergence and weak convergence in n-inner product spaces and study the relation between them. In particular, we show that the strong convergence implies the weak convergence and disprove the converse through a counter-example, by invoking an analogue of Parseval’s identity in n-inner product spaces.

We prove a multiparameter version of a classical theorem of Jones and Journé on weak-star convergence in the Hardy space H.