Three Equivalent n-Norms on the Space of p-Summable Sequences

Abstract

Given a normed space, one can define a new n-norm using a semi-inner product g on thespace, different from the n-norm defined by Gahler. In this paper, we are interested in the ¨new n-norm which is defined using such a functional g on the space $ell^p$ of p-summablesequences, where 1 ≤ p < ∞. We prove particularly that the new n-norm is equivalent withthe one defined previously by Gunawan on $ell^p$.