On the structure of duals in paranormed sequence spaces of quaternions

Abstract

This study develops a duality theory for paranormed sequence spaces whose terms take values in the non-commutative algebra of quaternions. Starting from Maddox-type positive exponent sequences, we introduce quaternion-valued analogues of the bounded, convergent, null and p-absolutely summable sequence classes. We then determine the corresponding a-, beta- and y-duals of these spaces. The main novelty of the study is that the duality problem is not treated as a direct scalar-field replacement of the classical real or complex case. Instead, the non-commutativity of quaternion multiplication is explicitly incorporated into the structure of the dual spaces. In particular, the convergence and boundedness conditions appearing in the definitions of the beta- and y-duals require a distinction between left and right multiplication, a phenomenon absent from classical sequence-space theory over commutative scalar fields. The obtained results extend Maddox-type paranormed sequence-space duality to the quaternionic setting and clarify the joint role of the paranorm topology and the order of quaternionic multiplication.