On the n-inner product spaces from the perspective of its quotient spaces

Abstract

In this paper, we investigate several topological properties of n-inner product spaces with respect to the inner products and norms defined on the quotient spaces we constructed. The construction is carried out with respect to a set of n linearly independent vectors, ensuring a consistent analytical framework. This construction was performed in several ways, each resulting in multiple quotient spaces. On each of these quotient spaces, we defined an inner product, along with the corresponding induced norm. Quotient spaces with similar structures are grouped into equivalence classes, thereby yielding several classes of quotient spaces. Within this framework, several topological aspects, including weak convergence, strong convergence, Cauchy sequences, and completeness, are examined with respect to classes of quotient spaces. Consequently, for each aspect, multiple definitions are formulated relative to these classes. We showed that the various definitions associated with a given topological property are equivalent to one another, regardless of the class of quotient spaces we used. Finally, the minimal number of norms within a given class required for an effective investigation of these properties is determined, thereby contributing to a more efficient and non-redundant analytical framework.