On the space of $p$-summable sequences

Abstract

On the space $\ell^p$ of $p$-summable sequences(of real numbers), one can derive a norm from the 2-norm asindicated by Gunawan [H. Gunawan, {The space of $p$-summablesequences and its natural $n$-norms}, Bull. Austral. Math.Soc. {64} (2001), 137–147]. The purpose of this noteis to establish the equivalence between such a norm andthe usual norm on $\ell^p$. We show that our result is usefulin understanding the topology of $\ell^p$ as a 2-normed space.