Abstract

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

Keywords: $p$-summable sequence space; norm equivalence.

MSC: 46B05, 46B20, 46A45, 46B45

Pages:  58$-$63     

Volume  65 ,  Issue  1 ,  2013